Debrecen, 2014

TÁMOP- 4.1.2.A/1-11/1-2011-0098

„Műszaki és gazdasági szakok alapozó matematikai

ismereteinek e–learning alapú tananyag- és módszertani fejlesztése”

College Geometry

Csaba Vincze and László Kozma

❈♦♥t❡♥ts

■♥tr♦❞✉❝t✐♦♥ ✾

■ ❊❧❡♠❡♥t❛r② ●❡♦♠❡tr② ✶✶

✶ ●❡♥❡r❛❧ ❝♦♠♣✉t❛t✐♦♥❛❧ s❦✐❧❧s ✶✸

✶✳✶ ◆✉♠❜❡rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✶✳✶✳✶ ◆❛t✉r❛❧ ♥✉♠❜❡rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✶✳✶✳✷ ■♥t❡❣❡rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✶✳✶✳✸ ❘❛t✐♦♥❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✶✳✶✳✹ ❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✶✳✶✳✺ ■rr❛t✐♦♥❛❧ ♥✉♠❜❡rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻✶✳✶✳✻ ❈♦♠♣❧❡① ♥✉♠❜❡rs✴✈❡❝t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻

✶✳✷ ❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻✶✳✸ ▲✐♠✐ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶

✶✳✸✳✶ ❆♣♣r♦①✐♠❛t✐♦♥ ♦❢ ✐rr❛t✐♦♥❛❧ ♥✉♠❜❡rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶✶✳✸✳✷ ❚❤❡ ♣r♦❜❧❡♠ ♦❢ t❤❡ s❤♦rt❡st ✇❛② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹✶✳✸✳✸ ❚❤❡ ❛r❡❛ ♦❢ t❤❡ ✉♥✐t ❝✐r❝❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺

✶✳✹ ❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻✶✳✺ ❋✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶

✶✳✺✳✶ ❊①♣♦♥❡♥t✐❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶✶✳✺✳✷ ❚r✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷✶✳✺✳✸ P♦❧②♥♦♠✐❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷

✶✳✻ ❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸✶✳✼ ▼❡❛♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶✶✳✽ ❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸✶✳✾ ❊q✉❛t✐♦♥s✱ s②st❡♠ ♦❢ ❡q✉❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹✶✳✶✵ ❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺

✷ ❊①❡r❝✐s❡s ✹✼

✷✳✶ ❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼

✸

✹ ❈❖◆❚❊◆❚❙

✸ ❇❛s✐❝ ❢❛❝ts ✐♥ ❣❡♦♠❡tr② ✺✼

✸✳✶ ❚❤❡ ❛①✐♦♠s ♦❢ ✐♥❝✐❞❡♥❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼✸✳✷ P❛r❛❧❧❡❧✐s♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽✸✳✸ ▼❡❛s✉r❡♠❡♥t ❛①✐♦♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾✸✳✹ ❈♦♥❣r✉❡♥❝❡ ❛①✐♦♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵✸✳✺ ❆r❡❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶✸✳✻ ❇❛s✐❝ ❢❛❝ts ✐♥ ❣❡♦♠❡tr② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶

✸✳✻✳✶ ❚r✐❛♥❣❧❡ ✐♥❡q✉❛❧✐t✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷✸✳✻✳✷ ❍♦✇ t♦ ❝♦♠♣❛r❡ tr✐❛♥❣❧❡s ■ ✲ ❝♦♥❣r✉❡♥❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷✸✳✻✳✸ ❈❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ♣❛r❛❧❧❡❧✐s♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹✸✳✻✳✹ ❍♦✇ t♦ ❝♦♠♣❛r❡ tr✐❛♥❣❧❡s ■■ ✲ s✐♠✐❧❛r✐t② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺

✹ ❚r✐❛♥❣❧❡s ✻✾

✹✳✶ ●❡♥❡r❛❧ tr✐❛♥❣❧❡s ■ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾✹✳✷ ❚❤❡ ❊✉❧❡r ❧✐♥❡ ❛♥❞ t❤❡ ❋❡✉❡r❜❛❝❤ ❝✐r❝❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶✹✳✸ ❙♣❡❝✐❛❧ tr✐❛♥❣❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹✹✳✹ ❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼✹✳✺ ❚r✐❣♦♥♦♠❡tr② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✾✹✳✻ ❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹✹✳✼ ●❡♥❡r❛❧ tr✐❛♥❣❧❡s ■■ ✲ ❙✐♥❡ ❛♥❞ ❈♦s✐♥❡ r✉❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✻

✹✳✼✳✶ ❙✐♥❡ r✉❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✻✹✳✼✳✷ ❈♦s✐♥❡ r✉❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✼✹✳✼✳✸ ❆r❡❛ ♦❢ tr✐❛♥❣❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✽

✹✳✽ ❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✵

✺ ❊①❡r❝✐s❡s ✾✸

✺✳✶ ❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✸

✻ ❈❧❛ss✐❝❛❧ ♣r♦❜❧❡♠s ■ ✶✵✶

✻✳✶ ❚❤❡ ♣r♦❜❧❡♠ ♦❢ t❤❡ t✉♥♥❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✶✻✳✷ ❍♦✇ t♦ ♠❡❛s✉r❡ ❛♥ ✉♥r❡❛❝❤❛❜❧❡ ❞✐st❛♥❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✷✻✳✸ ❍♦✇ ❢❛r ❛✇❛② ✐s t❤❡ ▼♦♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✸

✼ ◗✉❛❞r✐❧❛t❡r❛❧s ✶✵✼

✼✳✶ ●❡♥❡r❛❧ ♦❜s❡r✈❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✼✼✳✷ P❛r❛❧❧❡❧♦❣r❛♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✽✼✳✸ ❙♣❡❝✐❛❧ ❝❧❛ss❡s ♦❢ q✉❛❞r✐❧❛t❡r❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✵

✼✳✸✳✶ ❙②♠♠❡tr✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✶✼✳✸✳✷ ❆r❡❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✷

❈❖◆❚❊◆❚❙ ✺

✽ ❊①❡r❝✐s❡s ✶✶✺

✽✳✶ ❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✺

✾ P♦❧②❣♦♥s ✶✷✺

✾✳✶ P♦❧②❣♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✺

✶✵ ❈✐r❝❧❡s ✶✷✼

✶✵✳✶ ❚❛♥❣❡♥t ❧✐♥❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✼✶✵✳✷ ❚❛♥❣❡♥t✐❛❧ ❛♥❞ ❝②❝❧✐❝ q✉❛❞r✐❧❛t❡r❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✶✶✵✳✸ ❚❤❡ ❛r❡❛ ♦❢ ❝✐r❝❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✷

✶✶ ❊①❡r❝✐s❡s ✶✸✺

✶✶✳✶ ❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✺

✶✷ ●❡♦♠❡tr✐❝ tr❛♥s❢♦r♠❛t✐♦♥s ✶✹✺

✶✷✳✶ ■s♦♠❡tr✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✺✶✷✳✷ ❙✐♠✐❧❛r✐t✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✽

✶✸ ❈❧❛ss✐❝❛❧ ♣r♦❜❧❡♠s ■■ ✶✺✶

✶✸✳✶ ❚❤❡ ♣r♦❜❧❡♠ ♦❢ t❤❡ ❜r✐❞❣❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✶✶✸✳✷ ❚❤❡ ♣r♦❜❧❡♠ ♦❢ t❤❡ ❝❛♠❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✷✶✸✳✸ ❚❤❡ ❋❡r♠❛t ♣♦✐♥t ♦❢ ❛ tr✐❛♥❣❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✷

✶✹ ▲♦♥❣✐t✉❞❡s ❛♥❞ ❧❛t✐t✉❞❡s ✶✺✺

■■ ❆♥❛❧②t✐❝❛❧ ●❡♦♠❡tr② ✶✺✾

✶✺ ❈❛rt❡s✐❛♥ ❈♦♦r❞✐♥❛t❡s ✐♥ ❛ P❧❛♥❡ ✶✻✶

✶✺✳✶ ❈♦♦r❞✐♥❛t❡s ✐♥ ❛ ♣❧❛♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻✶✶✺✳✷ ❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻✺✶✺✳✸ ❚❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ♣♦✐♥ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻✻✶✺✳✹ ❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻✽✶✺✳✺ ❉✐✈✐❞✐♥❣ ❛ ❧✐♥❡ s❡❣♠❡♥t ✐♥ ❛ ❣✐✈❡♥ r❛t✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻✾✶✺✳✻ ❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼✶✶✺✳✼ ❚❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ❝✐r❝❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼✷✶✺✳✽ ❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼✹✶✺✳✾ ❚❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ❝✉r✈❡ r❡♣r❡s❡♥t❡❞ ❜② ♣❛r❛♠❡t❡rs ✳ ✳ ✳ ✳ ✳ ✳ ✶✼✺✶✺✳✶✵❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼✼

✻ ❈❖◆❚❊◆❚❙

✶✻ ❚❤❡ ❙tr❛✐❣❤t ▲✐♥❡ ✶✼✾

✶✻✳✶ ❚❤❡ ❣❡♥❡r❛❧ ❡q✉❛t✐♦♥ ♦❢ ❛ str❛✐❣❤t ❧✐♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼✾✶✻✳✷ P❛rt✐❝✉❧❛r ❝❛s❡s ♦❢ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ❧✐♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽✶✶✻✳✸ ❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽✸✶✻✳✹ ❚❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ t✇♦ str❛✐❣❤t ❧✐♥❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽✹✶✻✳✺ ❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽✻✶✻✳✻ ❚❤❡ ♣❛r❛❧❧❡❧✐s♠ ❛♥❞ ♣❡r♣❡♥❞✐❝✉❧❛r✐t② ♦❢ ❧✐♥❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽✼✶✻✳✼ ❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽✾✶✻✳✽ ❇❛s✐❝ ♣r♦❜❧❡♠s ♦♥ t❤❡ str❛✐❣❤t ❧✐♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽✾✶✻✳✾ ❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾✷

✶✼ ❱❡❝t♦rs ✶✾✸

✶✼✳✶ ❆❞❞✐t✐♦♥ ❛♥❞ s✉❜tr❛❝t✐♦♥ ♦❢ ✈❡❝t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾✸✶✼✳✷ ❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾✺✶✼✳✸ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ❜② ❛ ♥✉♠❜❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾✻✶✼✳✹ ❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾✽✶✼✳✺ ❙❝❛❧❛r ♣r♦❞✉❝t ♦❢ ✈❡❝t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾✾✶✼✳✻ ❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵✶✶✼✳✼ ❚❤❡ ✈❡❝t♦r ♣r♦❞✉❝t ♦❢ ✈❡❝t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵✶✶✼✳✽ ❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵✹✶✼✳✾ ❚❤❡ tr✐♣❧❡ ♣r♦❞✉❝t ♦❢ ✈❡❝t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵✹✶✼✳✶✵❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵✺

✶✽ ❘❡❝t❛♥❣✉❧❛r ❈❛rt❡s✐❛♥ ❈♦♦r❞✐♥❛t❡s ✐♥ ❙♣❛❝❡ ✷✵✼

✶✽✳✶ ❈❛rt❡s✐❛♥ ❝♦♦r❞✐♥❛t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵✼✶✽✳✷ ❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵✾✶✽✳✸ ❊❧❡♠❡♥t❛r② ♣r♦❜❧❡♠s ♦❢ s♦❧✐❞ ❛♥❛❧②t✐❝ ❣❡♦♠❡tr② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵✾✶✽✳✹ ❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶✶✶✽✳✺ ❊q✉❛t✐♦♥s ♦❢ ❛ s✉r❢❛❝❡ ❛♥❞ ❛ ❝✉r✈❡ ✐♥ s♣❛❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶✷

✶✾ ❆ P❧❛♥❡ ❛♥❞ ❛ ❙tr❛✐❣❤t ▲✐♥❡ ✷✶✺

✶✾✳✶ ❚❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ♣❧❛♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶✺✶✾✳✷ ❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶✻✶✾✳✸ ❙♣❡❝✐❛❧ ♣♦s✐t✐♦♥s ♦❢ ❛ ♣❧❛♥❡ r❡❧❛t✐✈❡ t♦ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ✳ ✳ ✳ ✷✶✽✶✾✳✹ ❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶✾✶✾✳✺ ❚❤❡ ♥♦r♠❛❧ ❢♦r♠ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ♣❧❛♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷✵✶✾✳✻ ❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷✶✶✾✳✼ ❘❡❧❛t✐✈❡ ♣♦s✐t✐♦♥ ♦❢ ♣❧❛♥❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷✷✶✾✳✽ ❊q✉❛t✐♦♥s ♦❢ t❤❡ str❛✐❣❤t ❧✐♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷✷✶✾✳✾ ❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷✹✶✾✳✶✵❇❛s✐❝ ♣r♦❜❧❡♠s ♦❢ str❛✐❣❤t ❧✐♥❡s ❛♥❞ ♣❧❛♥❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷✺

❈❖◆❚❊◆❚❙ ✼

✷✵ ❆❝❦♥♦✇❧❡❞❣❡♠❡♥t ✷✷✾

✽ ❈❖◆❚❊◆❚❙

■♥tr♦❞✉❝t✐♦♥

❚❤❡♦r② ♦r Pr❛❝t✐❝❡❄ ❇✉t ✇❤② ♦r❄ ❚❤❡♦r② ❛♥❞ Pr❛❝t✐❝❡✳ ❚❤✐s ✐st❤❡ ❆rs ▼❛t❤❡♠❛t✐❝❛✳

❆❧❢ré❞ ❘é♥②✐

❚❤❡ ✇♦r❞ ❣❡♦♠❡tr② ♠❡❛♥s ❡❛rt❤ ♠❡❛s✉r❡♠❡♥t✳ ❆s ❢❛r ❛s ✇❡ ❦♥♦✇ t❤❡❛♥❝✐❡♥t ❊❣②♣t✐❛♥s ✇❡r❡ t❤❡ ✜rst ♣❡♦♣❧❡ t♦ ❞♦ ❣❡♦♠❡tr② ❢r♦♠ ❛❜s♦❧✉t❡❧② ♣r❛❝✲t✐❝❛❧ ♣♦✐♥ts ♦❢ ✈✐❡✇✳ ❚❤❡ ❤✐st♦r✐❛♥ ❍❡r♦❞♦t✉s r❡❧❛t❡s t❤❛t ✐♥ ✶✸✵✵ ❇❈ ✧✐❢ ❛♠❛♥ ❧♦st ❛♥② ♦❢ ❤✐s ❧❛♥❞ ❜② t❤❡ ❛♥♥✉❛❧ ♦✈❡r✢♦✇ ♦❢ t❤❡ ◆✐❧❡ ❤❡ ❤❛❞ t♦ r❡♣♦rtt❤❡ ❧♦ss t♦ P❤❛r❛♦ ✇❤♦ ✇♦✉❧❞ t❤❡♥ s❡♥❞ ❛♥ ♦✈❡rs❡❡r t♦ ♠❡❛s✉r❡ t❤❡ ❧♦ss ❛♥❞♠❛❦❡ ❛ ♣r♦♣♦rt✐♦♥❛t❡ ❛❜❛t❡♠❡♥t ♦❢ t❤❡ t❛①✧ ❬✶❪✳ ❚❤❡ ●r❡❡❦s ✇❡r❡ t❤❡ ✜rstt♦ ♠❛❦❡ ♣r♦❣r❡ss ✐♥ ❣❡♦♠❡tr② ✐♥ t❤❡ s❡♥s❡ t❤❛t t❤❡② ♠❛❞❡ ✐t ❛❜str❛❝t✳ ❚❤❡②✐♥tr♦❞✉❝❡❞ t❤❡ ✐❞❡❛ ♦❢ ❝♦♥s✐❞❡r✐♥❣ ✐❞❡❛❧✐③❡❞ ♣♦✐♥ts ❛♥❞ ❧✐♥❡s✳ ❯s✐♥❣ P❧❛t♦✬s✇♦r❞s t❤❡ ♦❜❥❡❝ts ♦❢ ❣❡♦♠❡tr✐❝ ❦♥♦✇❧❡❞❣❡ ❛r❡ ❡t❡r♥❛❧✳ ❚❤❡ ●r❡❡❦ ❞❡❞✉❝t✐✈❡♠❡t❤♦❞ ❣✐✈❡s ❛ ❦✐♥❞ ♦❢ ❛♥s✇❡r t♦ t❤❡ q✉❡st✐♦♥ ❤♦✇ t♦ ♦❜t❛✐♥ ✐♥❢♦r♠❛t✐♦♥❛❜♦✉t t❤✐s ✐❞❡❛❧✐③❡❞ ✇♦r❧❞✳ ■t ✇❛s ❝♦❞✐✜❡❞ ❜② ❊✉❝❧✐❞ ❛r♦✉♥❞ ✸✵✵ ❇❈ ✐♥ ❤✐s❢❛♠♦✉s ❜♦♦❦ ❡♥t✐t❧❡❞ ❊❧❡♠❡♥ts ✇❤✐❝❤ ✐s ❛ s②st❡♠ ♦❢ ❝♦♥❝❧✉s✐♦♥s ♦♥ t❤❡ ❜❛s❡s♦❢ ✉♥q✉❡st✐♦♥❛❜❧❡ ♣r❡♠✐ss❡s ♦r ❛①✐♦♠s✳ ❚❤❡ ♠❡t❤♦❞ ♥❡❡❞s t✇♦ ❢✉♥❞❛♠❡♥t❛❧❝♦♥❝❡♣ts t♦ ❜❡❣✐♥ ✇♦r❦✐♥❣✿ ✉♥❞❡✜♥❡❞ t❡r♠s s✉❝❤ ❛s ♣♦✐♥ts✱ ❧✐♥❡s✱ ♣❧❛♥❡s ❡t❝✳❛♥❞ ❛①✐♦♠s ✭s♦♠❡t✐♠❡s t❤❡② ❛r❡ r❡❢❡rr❡❞ ❛s ♣r❡♠✐ss❡s ♦r ♣♦st✉❧❛t❡s✮ ✇❤✐❝❤❛r❡ t❤❡ ❜❛s✐❝ ❛ss✉♠♣t✐♦♥s ❛❜♦✉t t❤❡ t❡r♠s ♦❢ ❣❡♦♠❡tr②✳

❚❤❡ ♠❛t❡r✐❛❧ ❝♦❧❧❡❝t❡❞ ❤❡r❡ tr② t♦ ✜t t❤❡ ❞✐✛❡r❡♥t r❡q✉✐r❡♠❡♥ts ❝♦♠✐♥❣❢r♦♠ t❤❡ ❞✐✛❡r❡♥t tr❛❞✐t✐♦♥❛❧ ♣♦✐♥ts ♦❢ ✈✐❡✇✳ ❖♥❡ ♦❢ t❤❡♠ ✇❛♥ts t♦ s♦❧✈❡♣r♦❜❧❡♠s ✐♥ ♣r❛❝t✐❝❡✱ t❤❡ ♦t❤❡r ✇❛♥ts t♦ ❞❡✈❡❧♦♣ ❛♥ ❛❜str❛❝t t❤❡♦r② ✐♥❞❡♣❡♥✲❞❡♥t❧② ♦❢ t❤❡ ❡♠♣✐r✐❝❛❧ ✇♦r❧❞✳ ❆❧t❤♦✉❣❤ ✐t ✐s ❤❛r❞ t♦ r❡❛❧✐③❡ t❤❡ ❡q✉✐❧✐❜r✐✉♠♦❢ ❞✐✛❡r❡♥t r❡q✉✐r❡♠❡♥ts ✭❧❡❝t✉r❡ ✈s✳ s❡♠✐♥❛r ♦r t❤❡♦r② ✈s✳ ♣r❛❝t✐❝❡✮ ❆❧❢ré❞❘é♥②✐✬s ❆rs ▼❛t❤❡♠❛t✐❝❛ ❬✷❪ ❣✐✈❡s ✉s ❛ ♣❡r❢❡❝t st❛rt✐♥❣ ♣♦✐♥t✿ ❧❡❝t✉r❡s ❛♥❞s❡♠✐♥❛rs✱ t❤❡♦r② ❛♥❞ ♣r❛❝t✐❝❡✳

❚❤❡ ✜rst ❝❤❛♣t❡r ✐s ❞❡✈♦t❡❞ t♦ ❣❡♥❡r❛❧ ❝♦♠♣✉t❛t✐♦♥❛❧ s❦✐❧❧s r❡❧❛t❡❞ t♦♥✉♠❜❡rs✱ ❡q✉❛t✐♦♥s✱ s②st❡♠ ♦❢ ❡q✉❛t✐♦♥s✱ ❢✉♥❝t✐♦♥s ❡t❝✳ ❚❤❡s❡ t♦♦❧s ❛♥❞t❤❡ r❡❧❛t❡❞ ♠❡t❤♦❞s ❛r❡ ✇✐❞❡❧② ✉s❡❞ ✐♥ ♠❛t❤❡♠❛t✐❝s✳ ■♥ ❝❤❛♣t❡r ✸ ✇❡ ✐♠✲✐t❛t❡ t❤❡ ❞❡❞✉❝t✐✈❡ ♠❡t❤♦❞ ❜② ❝♦❧❧❡❝t✐♥❣ ❜❛s✐❝ ❢❛❝ts ✐♥ ❣❡♦♠❡tr②✳ ❙♦♠❡ ♦❢

✾

✶✵ ❈❖◆❚❊◆❚❙

t❤❡♠ ❛r❡ ❛①✐♦♠s ✐♥ t❤❡ str✐❝t s❡♥s❡ ♦❢ t❤❡ ✇♦r❞ s✉❝❤ ❛s t❤❡ ❛①✐♦♠s ♦❢ ✐♥✲❝✐❞❡♥❝❡✱ ♣❛r❛❧❧❡❧✐s♠✱ ♠❡❛s✉r❡♠❡♥t ❛①✐♦♠s ❛♥❞ ❝♦♥❣r✉❡♥❝❡ ❛①✐♦♠✳ ❲❡ ❤❛✈❡❛♥♦t❤❡r ❝♦❧❧❡❝t✐♦♥ ♦❢ ❢❛❝ts ✇❤✐❝❤ ❛r❡ ♥♦t ✭♦r ♥♦t ♥❡❝❡ss❛r✐❧②✮ ❛①✐♦♠s✳ ❚❤❡②❛r❡ ❢r❡q✉❡♥t❧② ✉s❡❞ ✐♥ ❣❡♦♠❡tr✐❝ ❛r❣✉♠❡♥t❛t✐♦♥s s✉❝❤ ❛s t❤❡ ♣❛r❛❧❧❡❧ ❧✐♥❡ ✐♥✲t❡rs❡❝t✐♥❣ t❤❡♦r❡♠ ♦r t❤❡ ❜❛s✐❝ ❝❛s❡s ♦❢ t❤❡ ❝♦♥❣r✉❡♥❝❡ ❛♥❞ t❤❡ s✐♠✐❧❛r✐t② ♦❢tr✐❛♥❣❧❡s✳ ■♥ s♦♠❡ ♦❢ t❤❡s❡ ❝❛s❡s t❤❡ ♣r♦♦❢ ✐s ❛✈❛✐❧❛❜❧❡ ❧❛t❡r ♦♥ ❛ ❤✐❣❤❡r st❛❣❡♦❢ t❤❡ t❤❡♦r②✳ ❈❤❛♣t❡r ✹ ✐s ❞❡✈♦t❡❞ t♦ t❤❡ ✐♥✈❡st✐❣❛t✐♦♥ ♦❢ tr✐❛♥❣❧❡s ✇❤✐❝❤❛r❡ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ ✜❣✉r❡s ✐♥ ❊✉❝❧✐❞❡❛♥ ❣❡♦♠❡tr② ❜❡❝❛✉s❡ q✉❛❞r✐❧❛t❡r❛❧s✭❝❤❛♣t❡r ✼✮ ♦r ♣♦❧②❣♦♥s ✭❝❤❛♣t❡r ✾✮ ❛r❡ ♠❛❞❡ ✉♣ ♦❢ ✜♥✐t❡❧② ♠❛♥② tr✐❛♥❣❧❡s❛♥❞ ♠♦st ♦❢ ♥♦t ♣♦❧②❣♦♥❛❧ s❤❛♣❡s ❧✐❦❡ ❝✐r❝❧❡s ✭❝❤❛♣t❡r ✶✵✮ ❝❛♥ ❜❡ ✐♠❛❣❡❞ ❛s❧✐♠✐ts ♦❢ ♣♦❧②❣♦♥s✳

❊❛❝❤ ❝❤❛♣t❡r ✐♥❝❧✉❞❡s ❡①❡r❝✐s❡s t♦♦✳ ▼♦st ♦❢ t❤❡♠ ❤❛✈❡ ❛ ❞❡t❛✐❧❡❞ s♦❧✉✲t✐♦♥✳ ❊①❡r❝✐s❡s ✐♥ s❡♣❛r❛t❡❞ ❝❤❛♣t❡rs ❣✐✈❡ ❛♥ ♦✈❡r✈✐❡✇ ❛❜♦✉t t❤❡ ♣r❡✈✐♦✉s❝❤❛♣t❡r✬s ♠❛t❡r✐❛❧✳ ❚❤❡ ❝❧❛ss✐❝❛❧ ♣r♦❜❧❡♠s ✭❝❤❛♣t❡r ✻ ❛♥❞ ❝❤❛♣t❡r ✶✸✮ ✐❧❧✉s✲tr❛t❡ ❤♦✇ t♦ ✉s❡ ❣❡♦♠❡tr② ✐♥ ♣r❛❝t✐❝❡✳ ❚❤❡② ❛❧s♦ ❤❛✈❡ ❛ ❤✐st♦r✐❝❛❧ ❝❤❛r❛❝t❡r❧✐❦❡ t❤❡ ♣r♦❜❧❡♠ ♦❢ t❤❡ t✉♥♥❡❧ ✭s❡❝t✐♦♥ ✻✳✶✮ ♦r ❤♦✇ ❢❛r ❛✇❛② ✐s t❤❡ ▼♦♦♥✭s❡❝t✐♦♥ ✻✳✸✮✳

❋✐❣✉r❡ ✶✿ ❆❧❢ré❞ ❘é♥②✐ ✭✶✾✷✶✲✶✾✼✵✮✳

P❛rt ■

❊❧❡♠❡♥t❛r② ●❡♦♠❡tr②

✶✶

❈❤❛♣t❡r ✶

●❡♥❡r❛❧ ❝♦♠♣✉t❛t✐♦♥❛❧ s❦✐❧❧s

✶✳✶ ◆✉♠❜❡rs

◆✉♠❜❡rs ❛r❡ ♦♥❡ ♦❢ t❤❡ ♠♦st t②♣✐❝❛❧ ♦❜❥❡❝ts ✐♥ ♠❛t❤❡♠❛t✐❝s✳

✶✳✶✳✶ ◆❛t✉r❛❧ ♥✉♠❜❡rs

❚♦ ❞❡✈❡❧♦♣ t❤❡ ♥♦t✐♦♥ ♦❢ ♥✉♠❜❡rs t❤❡ st❛rt✐♥❣ ♣♦✐♥t ✐s ❢♦r♠❡❞ ❜② t❤❡ s♦✲❝❛❧❧❡❞ ♥❛t✉r❛❧ ♥✉♠❜❡rs ❝❤❛r❛❝t❡r✐③❡❞ ❜② ❛ s❡t ♦❢ ❛①✐♦♠s ❞✉❡ t♦ t❤❡ ✶✾t❤❝❡♥t✉r② ■t❛❧✐❛♥ ♠❛t❤❡♠❛t✐❝✐❛♥ ●✉✐s❡♣♣❡ P❡❛♥♦✳ ❚❤❡ P❡❛♥♦ ❛①✐♦♠s ❞❡✜♥❡t❤❡ ❛r✐t❤♠❡t✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ ♥❛t✉r❛❧ ♥✉♠❜❡rs✱ ✉s✉❛❧❧② r❡♣r❡s❡♥t❡❞ ❛s ❛ s❡t

N = {(0), 1, 2, . . . , n, n+ 1, . . .}

❚❤❡ P❡❛♥♦✬s ❛①✐♦♠s ❛r❡ ❢♦r♠✉❧❛t❡❞ ❛s ❢♦❧❧♦✇s✳

P✶✳ ✶ ✐s ❛ ♥❛t✉r❛❧ ♥✉♠❜❡r ✭t❤❡ s❡t ♦❢ ♥❛t✉r❛❧ ♥✉♠❜❡rs ✐s ♥♦♥✲❡♠♣t②✮✳

❚❤❡ ♥❛t✉r❛❧s ❛r❡ ❛ss✉♠❡❞ t♦ ❜❡ ❝❧♦s❡❞ ✉♥❞❡r ❛ s✐♥❣❧❡✲✈❛❧✉❡❞ s✉❝❝❡ss♦r ✲❢✉♥❝t✐♦♥ ❙✭♥✮❂♥✰✶✳

P✷✳ ❙✭♥✮ ❜❡❧♦♥❣s t♦ ◆ ❢♦r ❡✈❡r② ♥❛t✉r❛❧ ♥✉♠❜❡r ♥✳

P❡❛♥♦✬s ♦r✐❣✐♥❛❧ ❢♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ❛①✐♦♠s ✉s❡❞ t❤❡ s②♠❜♦❧ ✶ ❢♦r t❤❡ ✧✜rst✧♥❛t✉r❛❧ ♥✉♠❜❡r ❛❧t❤♦✉❣❤ ❛①✐♦♠ P✶ ❞♦❡s ♥♦t ✐♥✈♦❧✈❡ ❛♥② s♣❡❝✐✜❝ ♣r♦♣❡rt✐❡s❢♦r t❤❡ ❡❧❡♠❡♥t ✶✳ ❚❤❡ ♥✉♠❜❡r ✷ ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❛s ✷❂❙✭✶✮ ❛♥❞ s♦ ♦♥✿ ✸❂❙✭✷✮✱✹❂❙✭✸✮✱ ✳✳✳ ❚❤❡ ♥❡①t t✇♦ ❛①✐♦♠s ❞❡✜♥❡ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤✐s r❡♣r❡s❡♥t❛t✐♦♥✳

P✸✳ ❚❤❡r❡ ✐s ♥♦ ❛♥② ♥❛t✉r❛❧ ♥✉♠❜❡r s❛t✐s❢②✐♥❣ ❙✭♥✮❂✶✳

P✹✳ ■❢ ❙✭♠✮❂❙✭♥✮ t❤❡♥ ♠❂♥✳

✶✸

✶✹ ❈❍❆P❚❊❘ ✶✳ ●❊◆❊❘❆▲ ❈❖▼P❯❚❆❚■❖◆❆▲ ❙❑■▲▲❙

❋✐❣✉r❡ ✶✳✶✿ ●r❛♣❤✐❝❛❧ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ✐♥t❡❣❡rs✳

❚❤❡s❡ ❛①✐♦♠s ✐♠♣❧② t❤❛t t❤❡ ❡❧❡♠❡♥ts ✶✱ ✷❂❙✭✶✮✱ ✸❂❙✭✷✮✱ ✳✳✳ ❛r❡ ❞✐st✐♥❝t♥❛t✉r❛❧ ♥✉♠❜❡rs ❜✉t ✇❡ ♥❡❡❞ t❤❡ s♦✲❝❛❧❧❡❞ ❛①✐♦♠ ♦❢ ✐♥❞✉❝t✐♦♥ t♦ ♣r♦✈✐❞❡t❤❛t t❤✐s ♣r♦❝❡❞✉r❡ ❣✐✈❡s ❛❧❧ ❡❧❡♠❡♥ts ♦❢ t❤❡ ♥❛t✉r❛❧s✳

P✺✳ ■❢ ❑ ✐s ❛ s❡t s✉❝❤ t❤❛t ✶ ✐s ✐♥ ❑ ❛♥❞ ❢♦r ❡✈❡r② ♥❛t✉r❛❧ ♥✉♠❜❡r ♥✱ ♥ ✐s ✐♥❑ ✐♠♣❧✐❡s t❤❛t ❙✭♥✮ ✐s ✐♥ ❑ t❤❡♥ ❑ ❝♦♥t❛✐♥s ❡✈❡r② ♥❛t✉r❛❧ ♥✉♠❜❡r✳

✶✳✶✳✷ ■♥t❡❣❡rs

❊q✉❛t✐♦♥ ✺✰①❂✷ ❤❛s ♥♦ ♥❛t✉r❛❧ s♦❧✉t✐♦♥s✳ ▲❡t ♠ ❛♥❞ ♥ ❜❡ ♥❛t✉r❛❧ ♥✉♠❜❡rs✳❊q✉❛t✐♦♥s ♦❢ t❤❡ ❢♦r♠

m+ x = n ✭✶✳✶✮

✇✐t❤♦✉t s♦❧✉t✐♦♥s ❛♠♦♥❣ ♥❛t✉r❛❧s ❧❡❛❞ ✉s t♦ ♥❡✇ q✉❛♥t✐t✐❡s ❝❛❧❧❡❞ ✐♥t❡❣❡rs✿

Z = {. . . ,−(n+ 1),−n, . . . ,−1, 0, 1, . . . , n, n+ 1, . . .}.

❆♥② ✐♥t❡❣❡r ❝♦rr❡s♣♦♥❞s t♦ ❛ ♣❛✐r ✭♠✱♥✮ ♦❢ ♥❛t✉r❛❧s ❜② ❡q✉❛t✐♦♥ ✶✳✶✳ ❚✇♦❡q✉❛t✐♦♥s ❛r❡ ❝❛❧❧❡❞ ❡q✉✐✈❛❧❡♥t ✐❢ t❤❡② ❤❛✈❡ ❡①❛❝t❧② t❤❡ s❛♠❡ s♦❧✉t✐♦♥s✳ ■❢ ✇❡❛❞❞ t❤❡ s✐❞❡s ♦❢ t❤❡ ❡q✉❛t✐♦♥s ♠✰①❂♥ ❛♥❞ ♥✬❂♠✬✰① t❤❡♥ ♠✬✰♥✰①❂♠✰♥✬✰①✳❚❤❡r❡❢♦r❡

m′ + n = m+ n′ ✭✶✳✷✮

✐s ❛ ❞✐r❡❝t ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ ❢♦r♠❛❧ ❡q✉✐✈❛❧❡♥❝❡✳ ❚❤❡ ♣❛✐rs ✭♠✱♥✮ ❛♥❞ ✭♠✬✱♥✬✮s❛t✐s❢②✐♥❣ ❡q✉❛t✐♦♥ ✶✳✷ r❡♣r❡s❡♥t t❤❡ s❛♠❡ ✐♥t❡❣❡r✳ ■♥ ❝❛s❡ ♦❢ ✭✺✱✷✮ t❤✐s ♥❡✇q✉❛♥t✐t② ✇✐❧❧ ❜❡ ✇r✐tt❡♥ ❛s ✲ ✸✳

✶✳✶✳ ◆❯▼❇❊❘❙ ✶✺

✶✳✶✳✸ ❘❛t✐♦♥❛❧s

❊q✉❛t✐♦♥ ✺①❂✷ ❤❛s ♥♦ ✐♥t❡❣❡r s♦❧✉t✐♦♥s✳ ▲❡t ♠6=✵ ❛♥❞ ♥ ❜❡ ✐♥t❡❣❡rs✳ ❊q✉❛✲t✐♦♥s ♦❢ t❤❡ ❢♦r♠

mx = n ✭✶✳✸✮

✇✐t❤♦✉t s♦❧✉t✐♦♥s ❛♠♦♥❣ ✐♥t❡❣❡rs ❧❡❛❞ ✉s t♦ ♥❡✇ q✉❛♥t✐t✐❡s ❝❛❧❧❡❞ r❛t✐♦♥❛❧s✿

Q = {n/m | n, m ∈ Z ❛♥❞ m 6= 0}.

❆♥② r❛t✐♦♥❛❧ ♥✉♠❜❡r ❝♦rr❡s♣♦♥❞s t♦ ❛ ♣❛✐r ✭♠✱♥✮ ♦❢ ✐♥t❡❣❡rs ❜② ❡q✉❛t✐♦♥ ✶✳✸✳❚✇♦ ❡q✉❛t✐♦♥s ❛r❡ ❝❛❧❧❡❞ ❡q✉✐✈❛❧❡♥t ✐❢ t❤❡② ❤❛✈❡ ❡①❛❝t❧② t❤❡ s❛♠❡ s♦❧✉t✐♦♥s✳■❢ ✇❡ ♠✉❧t✐♣❧② t❤❡ s✐❞❡s ♦❢ t❤❡ ❡q✉❛t✐♦♥s ♠①❂♥ ❛♥❞ ♥✬❂♠✬① t❤❡♥ ♠✬♥①❂♠♥✬①✳❚❤❡r❡❢♦r❡

m′n = mn′ ✭✶✳✹✮

✐s ❛ ❞✐r❡❝t ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ ❢♦r♠❛❧ ❡q✉✐✈❛❧❡♥❝❡✳ ❚❤❡ ♣❛✐rs ✭♠✱♥✮ ❛♥❞ ✭♠✬✱♥✬✮s❛t✐s❢②✐♥❣ ❡q✉❛t✐♦♥ ✶✳✹ r❡♣r❡s❡♥t t❤❡ s❛♠❡ r❛t✐♦♥❛❧ ♥✉♠❜❡r✳ ■♥ ❝❛s❡ ♦❢ ✭✺✱✷✮t❤✐s ♥❡✇ q✉❛♥t✐t② ✇✐❧❧ ❜❡ ✇r✐tt❡♥ ❛s ✷✴✺✳

✶✳✶✳✹ ❊①❡r❝✐s❡s

❊①❝❡r❝✐s❡ ✶✳✶✳✶ ❈❛❧❝✉❧❛t❡ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❞✐❛❣♦♥❛❧ ♦❢ ❛ sq✉❛r❡ ✇✐t❤ s✐❞❡♦❢ ✉♥✐t ❧❡♥❣t❤✳

❍✐♥t✳ ❯s✐♥❣ P②t❤❛❣♦r❡❛♥ t❤❡♦r❡♠ ✇❡ ❤❛✈❡ t❤❛t t❤❡ ❞✐❛❣♦♥❛❧ ✐s ❛ ♥✉♠❜❡rs❛t✐s❢②✐♥❣ ❡q✉❛t✐♦♥ x2 = 2✳

❊①❝❡r❝✐s❡ ✶✳✶✳✷ Pr♦✈❡ t❤❛t√2 ✐s ♥♦t ❛ r❛t✐♦♥❛❧ ♥✉♠❜❡r✳

❍✐♥t✳ ❙✉♣♣♦s❡ ✐♥ ❝♦♥tr❛r② t❤❛t

√2 =

n

m,

✇❤❡r❡ ♥ ❛♥❞ ♠ ❛r❡ ✐♥t❡❣❡rs✳ ❚❛❦✐♥❣ t❤❡ sq✉❛r❡ ♦❢ ❜♦t❤ s✐❞❡s ✇❡ ❤❛✈❡ t❤❛t

2m2 = n2,

✇❤❡r❡ t❤❡ ❧❡❢t ❤❛♥❞ s✐❞❡ ❝♦♥t❛✐♥s ❛♥ ♦❞❞ ♣♦✇❡r ♦❢ 2 ✐♥ t❤❡ ♣r✐♠❡ ❢❛❝t♦r✐③❛t✐♦♥✇❤✐❝❤ ❝♦♥tr❛❞✐❝ts t♦ t❤❡ ❡✈❡♥ ♣♦✇❡r ♦♥ t❤❡ r✐❣❤t ❤❛♥❞ s✐❞❡✳ ❚❤❡r❡❢♦r❡ t❤❡st❛rt✐♥❣ ❤②♣♦t❤❡s✐s ✐s ❢❛❧s❡✳

✶✻ ❈❍❆P❚❊❘ ✶✳ ●❊◆❊❘❆▲ ❈❖▼P❯❚❆❚■❖◆❆▲ ❙❑■▲▲❙

❋✐❣✉r❡ ✶✳✷✿ ❚❤❡ r♦♦ts♣✐r❛❧✳

✶✳✶✳✺ ■rr❛t✐♦♥❛❧ ♥✉♠❜❡rs

❉❡✜♥✐t✐♦♥ ◆✉♠❜❡rs ✇❤✐❝❤ ❝❛♥ ♥♦t ❜❡ ✇r✐tt❡♥ ❛s t❤❡ r❛t✐♦ ♦❢ ✐♥t❡❣❡rs ❛r❡❝❛❧❧❡❞ ✐rr❛t✐♦♥❛❧✳ ❚❤❡ s❡t ♦❢ r❡❛❧ ♥✉♠❜❡rs ❘ ❝♦♥s✐sts ♦❢ t❤❡ r❛t✐♦♥❛❧ ❛♥❞ t❤❡✐rr❛t✐♦♥❛❧ ♥✉♠❜❡rs✳

■rr❛t✐♦♥❛❧ ♥✉♠❜❡rs ❝❛♥ ❜❡ ✐♠❛❣❡❞ ❛s ❧✐♠✐ts ♦❢ s❡q✉❡♥❝❡s ♦❢ r❛t✐♦♥❛❧ ♥✉♠✲❜❡rs❀ s❡❡ s✉❜s❡❝t✐♦♥ ✶✳✸✳✶✳

✶✳✶✳✻ ❈♦♠♣❧❡① ♥✉♠❜❡rs✴✈❡❝t♦rs

❚♦ ❞❡✈❡❧♦♣ t❤❡ ♥♦t✐♦♥ ♦❢ ♥✉♠❜❡rs t❤❡ ♥❡①t ❧❡✈❡❧ ✐s t❤❡ ❝♦♠♣❧❡① ♥✉♠❜❡rs✇❤✐❝❤ ❝❛♥ ❜❡ ✐♥t❡r♣r❡t❡❞ ❛s ✈❡❝t♦rs ♦r ❡❧❡♠❡♥ts ✐♥ t❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡✳ ❚❤❡❛❧❣❡❜r❛✐❝ ♠♦t✐✈❛t✐♦♥ ✐s t♦ ♣r♦✈✐❞❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❡q✉❛t✐♦♥ x2 = −1✳

✶✳✷ ❊①❡r❝✐s❡s

■♥ ✇❤❛t ❢♦❧❧♦✇s ✇❡ s❤❛❧❧ ✉s❡ t❤❡ ♥♦t❛t✐♦♥ ♥✰✶ ✐♥st❡❛❞ ♦❢ ❙✭♥✮ ❢♦r t❤❡ s❛❦❡ ♦❢s✐♠♣❧✐❝✐t②✳

❊①❝❡r❝✐s❡ ✶✳✷✳✶ ❯s✐♥❣ ✐♥❞✉❝t✐♦♥ ♣r♦✈❡ t❤❛t

1 + 2 + . . .+ n =n(n+ 1)

2✭✶✳✺✮

❙♦❧✉t✐♦♥✳ ❲❡ ❝❛♥ ❝❤❡❝❦ ❞✐r❡❝t❧② t❤❛t ✐❢ ♥❂✶ t❤❡♥

1 =1(1 + 1)

2,

✶✳✷✳ ❊❳❊❘❈■❙❊❙ ✶✼

✐✳❡✳ ❡q✉❛t✐♦♥ ✶✳✺ ✐s tr✉❡✳ ❙✉♣♣♦s❡ t❤❛t ♥ s❛t✐s✜❡s ❡q✉❛t✐♦♥ ✶✳✺✱ ✐✳❡✳

1 + 2 + . . .+ n =n(n+ 1)

2✭✐♥❞✉❝t✐✈❡ ❤②♣♦t❤❡s✐s✮

❛♥❞ ♣r♦✈❡ t❤❛t

1 + 2 + . . .+ n+ (n+ 1) =(n+ 1)

(

(n+ 1) + 1)

2.

▲❡t ✉s st❛rt ❢r♦♠ t❤❡ ❧❡❢t ❤❛♥❞ s✐❞❡✳ ❯s✐♥❣ t❤❡ ✐♥❞✉❝t✐✈❡ ❤②♣♦t❤❡s✐s ✇❡ ❤❛✈❡t❤❛t

1 + 2 + . . .+ n+ (n+ 1) =n(n+ 1)

2+ (n+ 1) =

(n+ 1)(

(n+ 1) + 1)

2.

❚❤❡r❡❢♦r❡ ♥✰✶ ❛❧s♦ s❛t✐s✜❡s ❡q✉❛t✐♦♥ ✶✳✺✳ ❚❤❡ ✜♥❛❧ ❝♦♥❝❧✉s✐♦♥ ✐s t❤❛t t❤❡ s❡t♦❢ ♥❛t✉r❛❧ ♥✉♠❜❡rs s❛t✐s❢②✐♥❣ ❡q✉❛t✐♦♥ ✶✳✺ ❝♦✈❡rs ◆✳

❘❡♠❛r❦ ❆s ✇❡ ❝❛♥ s❡❡ ✐♥❞✉❝t✐♦♥ ✐s ❛ ✉s❡❢✉❧ ❣❡♥❡r❛❧ ♠❡t❤♦❞ t♦ ♣r♦✈❡ st❛t❡✲♠❡♥ts r❡❧❛t❡❞ t♦ ♥❛t✉r❛❧s✳ ❖♥❡ ♦❢ ✐ts ✇❡❛❦♥❡ss ✐s t❤❛t ✇❡ ❤❛✈❡ t♦ ❣✉❡ss ✇❤❛tt♦ ♣r♦✈❡✳

❊①❝❡r❝✐s❡ ✶✳✷✳✷ Pr♦✈❡ t❤❡ s♦✲❝❛❧❧❡❞ ●❛✉ss✐❛♥ ❢♦r♠✉❧❛ ✶✳✺ ✇✐t❤♦✉t ✐♥❞✉❝✲t✐♦♥✳

❙♦❧✉t✐♦♥✳ ▲❡tsn = 1 + 2 + . . .+ n

❜❡ t❤❡ ♣❛rt✐❛❧ s✉♠ ♦❢ t❤❡ ✜rst ♥ ♥❛t✉r❛❧ ♥✉♠❜❡r✳ ❚❛❦✐♥❣ t❤❡ s✉♠ ♦❢ ❡q✉❛t✐♦♥s

sn = 1 + 2 + . . .+ n

❛♥❞sn = n+ (n− 1) + . . .+ 1

✇❡ ❤❛✈❡ t❤❛t2sn = n(n+ 1)

❛♥❞ t❤❡ ●❛✉ss✐❛♥ ❢♦r♠✉❧❛ ❢♦❧❧♦✇s ✐♠♠❡❞✐❛t❡❧②✳

❊①❝❡r❝✐s❡ ✶✳✷✳✸ ❯s✐♥❣ ✐♥❞✉❝t✐♦♥ ♣r♦✈❡ t❤❛t

12 + 22 + 32 + . . .+ n2 =n(n+ 1)(2n+ 1)

6✭✶✳✻✮

✶✽ ❈❍❆P❚❊❘ ✶✳ ●❊◆❊❘❆▲ ❈❖▼P❯❚❆❚■❖◆❆▲ ❙❑■▲▲❙

❙♦❧✉t✐♦♥✳ ❋♦❧❧♦✇ t❤❡ st❡♣s ❛s ❛❜♦✈❡ t♦ ♣r♦✈❡ ❡q✉❛t✐♦♥ ✶✳✻✳ ■❢ ♥❂✶ t❤❡♥ ✇❡❝❛♥ ❡❛s✐❧② ❝❤❡❝❦ t❤❛t

12 =1(1 + 1)(2 · 1 + 1)

6,

✐✳❡✳ ❡q✉❛t✐♦♥ ✶✳✻ ✐s tr✉❡✳ ❙✉♣♣♦s❡ t❤❛t ♥ s❛t✐s✜❡s ❡q✉❛t✐♦♥ ✶✳✻✱ ✐✳❡✳

12 + 22 + 32 + . . .+ n2 =n(n+ 1)(2n+ 1)

6

❛♥❞ ♣r♦✈❡ t❤❛t

12 + 22 + 32 + . . .+ n2 + (n+ 1)2 =(n+ 1) ((n+ 1) + 1) (2(n+ 1) + 1)

6.

▲❡t ✉s st❛rt ❢r♦♠ t❤❡ ❧❡❢t ❤❛♥❞ s✐❞❡✳ ❯s✐♥❣ t❤❡ ✐♥❞✉❝t✐✈❡ ❤②♣♦t❤❡s✐s ✇❡ ❤❛✈❡t❤❛t

12 + 22 + 32 + . . .+ n2 + (n+ 1)2 =n(n+ 1)(2n+ 1)

6+ (n+ 1)2 =

=n(n+ 1)(2n+ 1) + 6(n+ 1)2

6=

(n+ 1)(

n(2n+ 1) + 6(n+ 1))

6=

(n+ 1)(2n2 + 7n+ 6)

6=

(n+ 1)(2n2 + 4n+ 3n+ 6)

6=

=(n+ 1)

(

2n(n+ 2) + 3(n+ 2))

6=

(n+ 1)(n+ 2)(2n+ 3)

6=

=(n+ 1)

(

(n+ 1) + 1)(

2(n+ 1) + 1)

6

❛s ✇❛s t♦ ❜❡ ♣r♦✈❡❞✳

❊①❝❡r❝✐s❡ ✶✳✷✳✹ ❯s✐♥❣ ✐♥❞✉❝t✐♦♥ ♣r♦✈❡ t❤❛t

3 | n3 + 5n+ 6. ✭✶✳✼✮

❙♦❧✉t✐♦♥✳ ■❢ ♥❂✶ t❤❡♥13 + 5 · 1 + 6 = 12

❛♥❞ 3 | 12✳ ❚❤❡ ❡①♣r❡ss✐♦♥ (n + 1)3 + 5(n + 1) + 6 ❝❛♥ ❜❡ ✇r✐tt❡♥ ✐♥t♦ t❤❡❢♦r♠

(n+ 1)3 + 5(n+ 1) + 6 = (n3 + 5n+ 6) + 3n2 + 3n+ 6,

✇❤❡r❡✱ ❜② t❤❡ ✐♥❞✉❝t✐✈❡ ❤②♣♦t❤❡s✐s✱ ❡❛❝❤ t❡r♠ ❝❛♥ ❜❡ ❞✐✈✐❞❡❞ ❜② ✸✳

✶✳✷✳ ❊❳❊❘❈■❙❊❙ ✶✾

❊①❝❡r❝✐s❡ ✶✳✷✳✺ Pr♦✈❡ t❤❛t t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❡q✉❛t✐♦♥s

x2 = 3, x2 = 5 ❛♥❞ x2 = 7

❛r❡ ✐rr❛t✐♦♥❛❧s✳

❙♦❧✉t✐♦♥✳ ▲❡t ♣ ❜❡ ❛♥ ❛r❜✐tr❛r② ♣r✐♠❡ ♥✉♠❜❡r ❛♥❞ s✉♣♣♦s❡ ✐♥ ❝♦♥tr❛r② t❤❛t

√p =

n

m

✇❤❡r❡ ♥ ❛♥❞ ♠ ❛r❡ ✐♥t❡❣❡rs✳ ❚❛❦✐♥❣ t❤❡ sq✉❛r❡ ♦❢ ❜♦t❤ s✐❞❡s ✇❡ ❤❛✈❡ t❤❛t

pm2 = n2,

✇❤❡r❡ t❤❡ ❧❡❢t ❤❛♥❞ s✐❞❡ ❝♦♥t❛✐♥s ❛♥ ♦❞❞ ♣♦✇❡r ♦❢ p ✐♥ t❤❡ ♣r✐♠❡ ❢❛❝t♦r✐③❛t✐♦♥✇❤✐❝❤ ❝♦♥tr❛❞✐❝ts t♦ t❤❡ ❡✈❡♥ ♣♦✇❡r ♦♥ t❤❡ r✐❣❤t ❤❛♥❞ s✐❞❡✳ ❚❤❡r❡❢♦r❡ t❤❡st❛rt✐♥❣ ❤②♣♦t❤❡s✐s ✐s ❢❛❧s❡✳

❊①❝❡r❝✐s❡ ✶✳✷✳✻ Pr♦✈❡ t❤❛t t❤❡ s✉♠ ❛♥❞ t❤❡ ❢r❛❝t✐♦♥ ♦❢ r❛t✐♦♥❛❧ ♥✉♠❜❡rs❛r❡ r❛t✐♦♥❛❧✳

❊①❝❡r❝✐s❡ ✶✳✷✳✼ ■s ✐t tr✉❡ ♦r ♥♦t❄ ❚❤❡ s✉♠ ♦❢ ❛ r❛t✐♦♥❛❧ ❛♥❞ ❛♥ ✐rr❛t✐♦♥❛❧♥✉♠❜❡r ✐s

• r❛t✐♦♥❛❧✳

• ✐rr❛t✐♦♥❛❧✳

❙♦❧✉t✐♦♥✳ ❯s✐♥❣ t❤❡ r❡s✉❧t ♦❢ t❤❡ ♣r❡✈✐♦✉s ❡①❡r❝✐s❡ t❤❡ ❛ss✉♠♣t✐♦♥√2 + 3 = r❛t✐♦♥❛❧

❣✐✈❡s ❛ ❝♦♥tr❛❞✐❝t✐♦♥✳ ❖♥❡ ❝❛♥ ❡❛s✐❧② ❣❡♥❡r❛❧✐③❡ t❤❡ ❛r❣✉♠❡♥t ❢♦r t❤❡ s✉♠ ♦❢❛♥② r❛t✐♦♥❛❧ ❛♥❞ ✐rr❛t✐♦♥❛❧ ♥✉♠❜❡r✳ ❚❤❡ ♠❡t❤♦❞ ✐s ❝❛❧❧❡❞ ✐♥❞✐r❡❝t ♣r♦♦❢✳

❊①❝❡r❝✐s❡ ✶✳✷✳✽ ❋✐♥❞ ✐rr❛t✐♦♥❛❧ ♥✉♠❜❡rs ❛ ❛♥❞ ❜ s✉❝❤ t❤❛t

• ❛✰❜ ✐s r❛t✐♦♥❛❧✱

• ❛✰❜ ✐s ✐rr❛t✐♦♥❛❧✱

• ❛✴❜ ✐s r❛t✐♦♥❛❧✱

• ❛✴❜ ✐s ✐rr❛t✐♦♥❛❧✳

✷✵ ❈❍❆P❚❊❘ ✶✳ ●❊◆❊❘❆▲ ❈❖▼P❯❚❆❚■❖◆❆▲ ❙❑■▲▲❙

❙♦❧✉t✐♦♥✳ ■❢a = 1−

√2 ❛♥❞ b =

√2

t❤❡♥ t❤❡ s✉♠ ♦❢ ❛✰❜ ✐s ♦❜✈✐♦✉s❧② r❛t✐♦♥❛❧✳ ▲❡t

a =√2 ❛♥❞ b =

√3.

■❢a+ b = r t❤❡♥ a = r − b

❛♥❞a2 = r2 − 2rb+ b2

✇❤✐❝❤ ♠❡❛♥s t❤❛t2 = r2 − 2r

√3 + 3,

✐✳❡✳ √3 =

r2 + 1

2r.

❚❤✐s ♠❡❛♥s t❤❛t r ❝❛♥ ♥♦t ❜❡ ❛ r❛t✐♦♥❛❧ ♥✉♠❜❡r✳ ■❢

a = b =√2

t❤❡♥ ✐ts r❛t✐♦ ✐s ♦❜✈✐♦✉s❧② r❛t✐♦♥❛❧✳ ❋✐♥❛❧❧②✱ ✐❢

a =√2 + 1 ❛♥❞ b =

√2− 1

t❤❡♥

a/b =

√2 + 1√2− 1

=

√2 + 1√2− 1

·√2 + 1√2 + 1

=

3 + 2√2

1= 3 + 2

√2

✇❤✐❝❤ ✐s ♦❜✈✐♦✉s❧② ✐rr❛t✐♦♥❛❧✳

❊①❝❡r❝✐s❡ ✶✳✷✳✾ Pr♦✈❡ t❤❛t ①❂✷✰✸✐ s❛t✐s✜❡s ❡q✉❛t✐♦♥

x2 − 4x+ 13 = 0

❙♦❧✉t✐♦♥✳ ❙✐♥❝❡ t❤❡ ✐♠❛❣✐♥❛r② ✉♥✐t ✐s t❤❡ ❢♦r♠❛❧ s♦❧✉t✐♦♥ ♦❢ ❡q✉❛t✐♦♥ x2 = −1✇❡ ❤❛✈❡ t❤❛t

(2 + 3i)2 − 4(2 + 3i) + 13 = 4 + 12i+ 9i2 − 8− 12i+ 13 =

4 + 12i− 9− 8− 12i+ 13 = 0

✉s✐♥❣ t❤❡ ♣r✐♥❝✐♣❧❡ ♦❢ ♣❡r♠❛♥❡♥❝❡✿ ❦❡❡♣ ❛❧❧ t❤❡ ❛❧❣❡❜r❛✐❝ r✉❧❡s ♦❢ ❝❛❧❝✉❧❛t✐♦♥✇✐t❤ r❡❛❧s✳

✶✳✸✳ ▲■▼■❚❙ ✷✶

✶✳✸ ▲✐♠✐ts

■♥ t❤✐s s❡❝t✐♦♥ ✇❡ ✐❧❧✉str❛t❡ ❤♦✇ ✐rr❛t✐♦♥❛❧ ♥✉♠❜❡rs ❝❛♥ ❜❡ ✐♥t❡r♣r❡t❡❞ ❛s❧✐♠✐ts ♦❢ s❡q✉❡♥❝❡s ♦❢ r❛t✐♦♥❛❧ ♥✉♠❜❡rs✳ ❚❛❦✐♥❣ t❤❡ ❧✐♠✐t ✐s ♦♥❡ ♦❢ t❤❡ ♠♦st✐♠♣♦rt❛♥t ♦♣❡r❛t✐♦♥s ✐♥ ♠❛t❤❡♠❛t✐❝s✳ ■t ✐s ✉s❡❞ ✐♥ t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ t❤❡♥♦t✐♦♥ ♦❢ ♥✉♠❜❡rs✱ t❤❡ t❤❡♦r② ♦❢ ❧❡♥❣t❤✱ ❛r❡❛ ❛♥❞ ✈♦❧✉♠❡ ♦❢ ❣❡♥❡r❛❧ s❤❛♣❡s✭❝✉r✈❡s✱ s✉r❢❛❝❡s ❛♥❞ ❜♦❞✐❡s✮ ❛♥❞ s♦ ♦♥✳ ❍❡r❡ ✇❡ ❛♣♣❧② ♦♥❧② ❛ ❦✐♥❞ ♦❢ ✐♥t✉✐t✐♦♥t♦ ❝r❡❛t❡ ❧✐♠✐ts ✇✐t❤♦✉t ♣r❡❝✐s❡ ❞❡✜♥✐t✐♦♥s✳

✶✳✸✳✶ ❆♣♣r♦①✐♠❛t✐♦♥ ♦❢ ✐rr❛t✐♦♥❛❧ ♥✉♠❜❡rs

■t ❝❛♥ ❜❡ ❡❛s✐❧② s❡❡♥ t❤❛t1 <

√2 < 2.

❈♦♥s✐❞❡r t❤❡ ♠✐❞♣♦✐♥t

q1 =1 + 2

2= 3/2

♦❢ t❤❡ ✐♥t❡r✈❛❧ [1, 2]✳ ❚❛❦✐♥❣ t❤❡ sq✉❛r❡ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s✐❞❡s ✐t ❝❛♥ ❜❡♣r♦✈❡❞ t❤❛t

1 <√2 < 3/2

❛♥❞ ✇❡ ❤❛✈❡ ❛ ❜❡tt❡r ❛♣♣r♦①✐♠❛t✐♦♥ ❜② t❤❡ ♠✐❞♣♦✐♥t

q2 :=1 + (3/2)

2= 5/4.

❚❛❦✐♥❣ t❤❡ sq✉❛r❡ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s✐❞❡s ❛❣❛✐♥ ✐t ❝❛♥ ❜❡ ♣r♦✈❡❞ t❤❛t

5/4 <√2 < 3/2

❛♥❞ t❤❡ ♠✐❞♣♦✐♥t

q3 =(5/4) + (3/2)

2= 11/8

✐s ❛ ❜❡tt❡r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢√2✳ ❚❤❡ ♠❡t❤♦❞ ✐s s✐♠✐❧❛r t♦ ❧♦♦❦✐♥❣ ❢♦r ❛ ✇♦r❞

✐♥ ❛ ❞✐❝t✐♦♥❛r②✳ ❚❤❡ ❜❛s✐❝ st❡♣s ❛r❡

• ♦♣❡♥ t❤❡ ❞✐❝t✐♦♥❛r② ✐♥ ❛ r❛♥❞♦♠ ✇❛② ✭❢♦r ❡①❛♠♣❧❡ ♦♣❡♥ t❤❡ ❜♦♦❦ ✐♥t❤❡ ♠✐❞❞❧❡ ♣❛rt✮

• ❝♦♠♣❛r❡ t❤❡ ✇♦r❞ ✇❡ ❛r❡ ❧♦♦❦✐♥❣ ❢♦r ✇✐t❤ t❤❡ ✐♥✐t✐❛❧ ❧❡tt❡r ♦❢ t❤❡ ✇♦r❞s♦♥ t❤❡ s❤❡❡t✳

✷✷ ❈❍❆P❚❊❘ ✶✳ ●❊◆❊❘❆▲ ❈❖▼P❯❚❆❚■❖◆❆▲ ❙❑■▲▲❙

❋✐❣✉r❡ ✶✳✸✿ ❆♣♣r♦①✐♠❛t✐♦♥ ♦❢ sq✉❛r❡ r♦♦t ✷✳

❊✈❡r② t✐♠❡ ✇❡ ❜✐s❡❝t t❤❡ ❞✐❝t✐♦♥❛r② ❜❡❢♦r❡ r✉♥♥✐♥❣ t❤❡ ❛❧❣♦r✐t❤♠ ❛❣❛✐♥✳ ❚❤❡♣r♦❝❡ss ✐s ♥♦t ❡①❛❝t❧② t❤❡ s❛♠❡ ❜✉t ✇❡ ✉s❡ t❤❡ s❛♠❡ ♣❤✐❧♦s♦♣❤② t♦ s♦❧✈❡ t❤❡♣r♦❜❧❡♠ ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢

√2✳ ❚❤❡ ♠♦st ❡ss❡♥t✐❛❧ ❞✐✛❡r❡♥❝❡ ✐s t❤❛t t❤❡

♠❡t❤♦❞ ♦❢ ✜♥❞✐♥❣√2 ✐s ♥♦t ✜♥✐t❡✿ ✇❡ ❛❧✇❛②s ❤❛✈❡ r❛t✐♦♥❛❧ ♥✉♠❜❡rs ✇❤✐❝❤

♠❡❛♥s t❤❛t ✇❡ ❝♦✉❧❞ ♥♦t ✜♥❞ t❤❡ ❡①❛❝t ✈❛❧✉❡ ♦❢√2 ❛♠♦♥❣ t❤❡ ♠❡♠❜❡rs ♦❢

t❤❡ s❡q✉❡♥❝❡ q1✱ q2✱ q3✱ ✳✳✳ ❇✉t t❤❡ ❡rr♦rs ❝❛♥ ❜❡ ❡st✐♠❛t❡❞ ❜② ❞❡❝r❡❛s✐♥❣✈❛❧✉❡s ❛s ❢♦❧❧♦✇s✿

|√2− q1| < t❤❡ ❤❛❧❢ ♦❢ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ✐♥t❡r✈❛❧ [1, 2] =

1

2=

1

21.

|√2− q2| < t❤❡ ❤❛❧❢ ♦❢ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ✐♥t❡r✈❛❧ [1, 3/2] =

1

4=

1

22.

■♥ ❛ s✐♠✐❧❛r ✇❛②

|√2− q3| < t❤❡ ❤❛❧❢ ♦❢ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ✐♥t❡r✈❛❧ [5/4, 3/2] =

1

8=

1

23.

■♥ ❣❡♥❡r❛❧

|√2− qn| <

1

2n.

❚❤❡r❡❢♦r❡ ✇❡ ❝❛♥ ❜❡ ❛s ❝❧♦s❡ t♦√2 ❛s ✇❡ ✇❛♥t t♦✳ ■♥ ♦t❤❡r ✇♦r❞s t❤❡ s❡q✉❡♥❝❡

q1✱ q2✱ q3✱ ✳✳✳ t❡♥❞s t♦√2 ❛♥❞ t❤✐s ♥✉♠❜❡r ❝❛♥ ❜❡ ✐♥t❡r♣r❡t❡❞ ❛s t❤❡ ❧✐♠✐t ♦❢

❛ s❡q✉❡♥❝❡ ♦❢ r❛t✐♦♥❛❧ ♥✉♠❜❡rs✳

❘❡♠❛r❦ ■♥ ✇❤❛t ❢♦❧❧♦✇s ✇❡ ♣r❡s❡♥t ❛ ▼❆P▲❊ ♣r♦❝❡❞✉r❡ ❢♦r t❤❡ ❛♣♣r♦①✐✲♠❛t✐♦♥ ♦❢ t❤❡ sq✉❛r❡ r♦♦t ♦❢ ♥❛t✉r❛❧s ❛s ✇❡ ❤❛✈❡ s❡❡♥ ❛❜♦✈❡✿ ❧❡t k ❜❡ ❛ ❣✐✈❡♥♥❛t✉r❛❧ ♥✉♠❜❡r✳ ❲❡ ❛r❡ ❣♦✐♥❣ t♦ ❛♣♣r♦①✐♠❛t❡ t❤❡ sq✉❛r❡ r♦♦t ♦❢ k ❜② ✉s✐♥❣t❤❡ ❜❛s✐❝ st❡♣ n t✐♠❡s✳ ❚❤❡ ♥❛♠❡ ♦❢ t❤❡ ♣r♦❝❡❞✉r❡ ✐s

f := ♣r♦❝(n, k)

❆t ✜rst ✇❡ s❤♦✉❧❞ ✜♥❞ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s

a <√k < b

✶✳✸✳ ▲■▼■❚❙ ✷✸

❛s ❢♦❧❧♦✇s✿a := 1;

b := 1;

✇❤✐❧❡ a2 < k ❞♦

a := a+ 1;

❡♥❞ ❞♦;

a := a− 1;

❚❤✐s ♠❡❛♥s t❤❛t ✐❢ t❤❡ ❛❝t✉❛❧ ✈❛❧✉❡ ♦❢ t❤❡ ✈❛r✐❛❜❧❡ a s❛t✐s✜❡s t❤❡ ✐♥❡q✉❛❧✐t②a2 < k t❤❡♥ ✇❡ ✐♥❝r❡❛s❡ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ✈❛r✐❛❜❧❡ ❜② ❛❞❞✐♥❣ ♦♥❡ ❛s ❢❛r ❛s♣♦ss✐❜❧❡✳ ❋✐♥❛❧❧② ✧❛✧ t❛❦❡s t❤❡ ❧❛st ✈❛❧✉❡ ❢♦r ✇❤✐❝❤ t❤❡ ✐♥❡q✉❛❧✐t② a2 < k ✐str✉❡✳ ❚❤❡ ✉♣♣❡r ❜♦✉♥❞ ✐s ❝r❡❛t❡❞ ✐♥ ❛ s✐♠✐❧❛r ✇❛②✿

✇❤✐❧❡ b2 < k ❞♦

b := b+ 1;

❡♥❞ ❞♦;

❆s t❤❡ ♥❡①t st❡♣ ✇❡ ❣✐✈❡ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♦❢ ❛ ♥❡✇ ✈❛r✐❛❜❧❡

c :=a+ b

2;

❛♥❞ ✇❡ ✉s❡ ❛ ✧❢♦r✧ ❧♦♦♣ t♦ t❛❦❡ t❤❡ ❤❛❧❢ ♦❢ t❤❡ ❡♥❝❧♦s✐♥❣ ✐♥t❡r✈❛❧❧s ♥ t✐♠❡s✿

❢♦r i ❢r♦♠ 1t♦ n ❞♦

✐❢ c2 < k t❤❡♥

a := c;

c :=a+ b

2;

❡❧s❡

b := c;

c :=a+ b

2;

❡♥❞ ✐❢;

❡♥❞ ❞♦;

r❡t✉r♥(c)

❡♥❞ ♣r♦❝;

❚❤❡ ✜❣✉r❡ s❤♦✇s ❤♦✇ t❤❡ ♣r♦❝❡❞✉r❡ ✐s ✇♦r❦✐♥❣ ✐♥ ❛ st❛♥❞❛r❞ ▼❛♣❧❡ ✇♦r❦s❤❡❡t❡♥✈✐r♦♥♠❡♥t✳

✷✹ ❈❍❆P❚❊❘ ✶✳ ●❊◆❊❘❆▲ ❈❖▼P❯❚❆❚■❖◆❆▲ ❙❑■▲▲❙

❋✐❣✉r❡ ✶✳✹✿ ❆ ▼❆P▲❊ ♣r♦❝❡❞✉r❡✳

✶✳✸✳✷ ❚❤❡ ♣r♦❜❧❡♠ ♦❢ t❤❡ s❤♦rt❡st ✇❛②

❖♥❡ ♦❢ t❤❡ ♠♦st ✐♠♣♦rt❛♥t ❜❛s✐❝ ❢❛❝t ✐♥ ❣❡♦♠❡tr② ✐s t❤❡ s♦✲❝❛❧❧❡❞ tr✐❛♥❣❧❡✐♥❡q✉❛❧✐t②

AC ≤ AB +BC ✭✶✳✽✮

t♦ ❡①♣r❡ss ❛ ♠♦r❡ ❣❡♥❡r❛❧ ❣❡♦♠❡tr✐❝ ♣r✐♥❝✐♣❧❡✳ ■t s❛②s t❤❛t t❤❡ s❤♦rt❡st ✇❛②❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ✐s t❤❡ str❛✐❣❤t ❧✐♥❡✳ ❚❤❡ q✉❡st✐♦♥ ✐s ❤♦✇ t♦ ❞❡r✐✈❡ t❤✐s♣r✐♥❝✐♣❧❡ ❢r♦♠ ✐♥❡q✉❛❧✐t② ✶✳✽ ✐♥ ❣❡♥❡r❛❧✳ ❚❤❡ ✜rst st❡♣ ✐s t❤❡ ❣❡♥❡r❛❧✐③❛t✐♦♥♦❢ t❤❡ tr✐❛♥❣❧❡ ✐♥❡q✉❛❧✐t②✳ ❯s✐♥❣ ❛ s✐♠♣❧❡ ✐♥❞✉❝t✐♦♥ ✇❡ ❝❛♥ ♣r♦✈❡ ♣♦❧②❣♦♥❛❧✐♥❡q✉❛❧✐t✐❡s

AC ≤ AB1 +B1B2 +B2C, AC ≤ AB1 +B1B2 +B2B3 +B3C ❛♥❞ s♦ ♦♥.

■♥ ❣❡♥❡r❛❧

AC ≤ AB1 +B1B2 +B2B3 + . . .+Bn−1Bn +BnC ✭✶✳✾✮

❢♦r ❛♥② ♥❛t✉r❛❧ ♥✉♠❜❡r n ≥ 3✳ ◆♦✇ ✐♠❛❣❡ ❛♥ ✧❛r❝✧ ❢r♦♠ ❆ t♦ ❈✳ ■❢ t❤❡❛r❝❧❡♥❣t❤ ✐s ✉♥❞❡rst♦♦❞ ❛s t❤❡ ❧✐♠✐t ♦❢ ❧❡♥❣t❤s ♦❢ ✐♥s❝r✐❜❡❞ ♣♦❧②❣♦♥❛❧ ❝❤❛✐♥s✐♥ s♦♠❡ s❡♥s❡ t❤❡♥ ✇❡ ❤❛✈❡ t❤❛t t❤❡ s❤♦rt❡st ✇❛② ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ✐s t❤❡str❛✐❣❤t ❧✐♥❡✳

✶✳✸✳ ▲■▼■❚❙ ✷✺

✶✳✸✳✸ ❚❤❡ ❛r❡❛ ♦❢ t❤❡ ✉♥✐t ❝✐r❝❧❡

❊✈❡r②❜♦❞② ❦♥♦✇s t❤❛t t❤❡ ❛r❡❛ ♦❢ ❛ ❝✐r❝❧❡ ✇✐t❤ r❛❞✐✉s r ✐s r2π✳ ■❢ ✇❡ ❤❛✈❡ ❛✉♥✐t ❝✐r❝❧❡ t❤❡♥ t❤❡ ❛r❡❛ ✐s ❥✉st π✳ ❍♦✇ ❝❛♥ ✇❡ ❝❛❧❝✉❧❛t❡ t❤❡ ✈❛❧✉❡ ♦❢ π❄

❚❤❡ ❡❛r❧✐❡st ❦♥♦✇♥ t❡①t✉❛❧❧② ❡✈✐❞❡♥❝❡❞ ❛♣♣r♦①✐♠❛t✐♦♥s ♦❢ π ❛r❡ ❢r♦♠❛r♦✉♥❞ ✶✾✵✵ ❇❈✳ ❚❤❡② ❛r❡ ❢♦✉♥❞ ✐♥ t❤❡ ❊❣②♣t✐❛♥ ❘❤✐♥❞ P❛♣②r✉s

π ≈ 256/81

❛♥❞ ♦♥ ❇❛❜②❧♦♥✐❛♥ t❛❜❧❡tsπ ≈ 25/8.

❚❤❡ ■♥❞✐❛♥ t❡①t ❙❤❛t❛♣❛t❤❛ ❇r❛❤♠❛♥❛ ❣✐✈❡s π ❛s ✸✸✾✴✶✵✽✳ ❆r❝❤✐♠❡❞❡s✭✷✽✼ ✲ ✷✶✷ ❇❈✮ ✇❛s t❤❡ ✜rst t♦ ❡st✐♠❛t❡ π r✐❣♦r♦✉s❧②✳ ❍❡ r❡❛❧✐③❡❞ t❤❛t ✐ts♠❛❣♥✐t✉❞❡ ❝❛♥ ❜❡ ❜♦✉♥❞❡❞ ❢r♦♠ ❜❡❧♦✇ ❛♥❞ ❛❜♦✈❡ ❜② t❤❡ ❛r❡❛ ♦❢ ✐♥s❝r✐❜✐♥❣❛♥❞ ❝✐r❝✉♠s❝r✐❜✐♥❣ r❡❣✉❧❛r ♣♦❧②❣♦♥s✳ ❋♦r ❡①❛♠♣❧❡ ✇❡ ❝❛♥ ✐♥s❝r✐❜❡ ✐♥ t❤❡❝✐r❝❧❡ ❛ r❡❣✉❧❛r ❤❡①❛❣♦♥ ♠❛❞❡ ✉♣ ♦❢ s✐① ❞✐s❥♦✐♥t ❡q✉✐❧❛t❡r❛❧ tr✐❛♥❣❧❡s ♦❢ s✐❞❡✶✳ ❚❤❡ ❛r❡❛ ♦❢ ❡❛❝❤ tr✐❛♥❣❧❡ ✐s ✸✴✭✹

√3✮ ❜② ❍ér♦♥✬s ❢♦r♠✉❧❛✱ s♦ t❤❡ ❛r❡❛ ♦❢

t❤❡ ❤❡①❛❣♦♥ ✐s

63

4√3=

9

2√3≈ 2.59808.

❚❤❡ ❛r❡❛ ♦❢ t❤❡ ❝✐r❝❧❡ s❤♦✉❧❞ ❜❡ ♦❜✈✐♦✉s❧② ❣r❡❛t❡r t❤❛♥ t❤✐s ✈❛❧✉❡✳ ■❢ ✇❡❝✐r❝✉♠s❝r✐❜❡ ❛ r❡❣✉❧❛r ❤❡①❛❣♦♥ ❛r♦✉♥❞ t❤❡ ✉♥✐t ❝✐r❝❧❡ t❤❡♥ t❤❡ ❛r❡❛ ❝❛♥ ❜❡❡st✐♠❛t❡❞ ❢r♦♠ ❛❜♦✈❡✳ ❚❤❡ ❛r❡❛ ♦❢ t❤❡ ❝✐r❝❧❡ s❤♦✉❧❞ ❜❡ ♦❜✈✐♦✉s❧② ❧❡ss t❤❛♥t❤❡ ❛r❡❛ ♦❢ t❤❡ ❝✐r❝✉♠s❝r✐❜❡❞ r❡❣✉❧❛r ❤❡①❛❣♦♥ ♦❢ s✐❞❡ ✷✴

√3✿

t❤❡ ❛r❡❛ ♦❢ t❤❡ ✉♥✐t ❝✐r❝❧❡ ≤ 61√3≈ 3.46410

❛♥❞ s♦ ♦♥✳ ❆r♦✉♥❞ ✹✽✵ ❩✉ ❈❤♦♥❣③❤✐ ❞❡♠♦♥str❛t❡❞ t❤❛t π ≈ 355/113 =3, 1415929✳ ❍❡ ❛❧s♦ s❤♦✇❡❞ t❤❛t 3, 1415926 < π < 3, 1415927✳

❚❤❡ ♥❡①t ♠❛❥♦r ❛❞✈❛♥❝❡s ✐♥ t❤❡ st✉❞② ♦❢ π ❝❛♠❡ ✇✐t❤ t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢✐♥✜♥✐t❡ s❡r✐❡s ❛♥❞ s✉❜s❡q✉❡♥t❧② ✇✐t❤ t❤❡ ❞✐s❝♦✈❡r② ♦❢ ❝❛❧❝✉❧✉s✴❛♥❛❧②s✐s✱ ✇❤✐❝❤♣❡r♠✐t t❤❡ ❡st✐♠❛t✐♦♥ ♦❢ π t♦ ❛♥② ❞❡s✐r❡❞ ❛❝❝✉r❛❝② ❜② ❝♦♥s✐❞❡r✐♥❣ s✉✣❝✐❡♥t❧②♠❛♥② t❡r♠s ♦❢ ❛ r❡❧❡✈❛♥t s❡r✐❡s✳ ❆r♦✉♥❞ ✶✹✵✵✱ ▼❛❞❤❛✈❛ ♦❢ ❙❛♥❣❛♠❛❣r❛♠❛❢♦✉♥❞ t❤❡ ✜rst ❦♥♦✇♥ s✉❝❤ s❡r✐❡s✿

π =4

1− 4

3+

4

5− 4

7+

4

9− 4

11+ · · · .

❚❤✐s ✐s ❦♥♦✇♥ ❛s t❤❡ ▼❛❞❤❛✈❛✲▲❡✐❜♥✐③ s❡r✐❡s ♦r ●r❡❣♦r②✲▲❡✐❜♥✐③ s❡r✐❡s s✐♥❝❡ ✐t✇❛s r❡❞✐s❝♦✈❡r❡❞ ❜② ❏❛♠❡s ●r❡❣♦r② ❛♥❞ ●♦tt❢r✐❡❞ ▲❡✐❜♥✐③ ✐♥ t❤❡ ✶✼t❤ ❝❡♥t✉r②✳▼❛❞❤❛✈❛ ✇❛s ❛❜❧❡ t♦ ❡st✐♠❛t❡ t❤❡ ✈❛❧✉❡ ♦❢ π ❝♦rr❡❝t❧② t♦ ✶✶ ❞❡❝✐♠❛❧ ♣❧❛❝❡s✳

✷✻ ❈❍❆P❚❊❘ ✶✳ ●❊◆❊❘❆▲ ❈❖▼P❯❚❆❚■❖◆❆▲ ❙❑■▲▲❙

❚❤❡ r❡❝♦r❞ ✇❛s ❜❡❛t❡♥ ✐♥ ✶✹✷✹ ❜② t❤❡ P❡rs✐❛♥ ♠❛t❤❡♠❛t✐❝✐❛♥✱ ❏❛♠s❤✐❞ ❛❧✲❑❛s❤✐ ❜② ❣✐✈✐♥❣ ❛♥ ❡st✐♠❛t✐♦♥ t❤❛t ✐s ❝♦rr❡❝t t♦ ✶✻ ❞❡❝✐♠❛❧ ❞✐❣✐ts✳ ❚❤❡❛❝❝✉r❛❝② ✉♣ t♦ ✸✺ ❞❡❝✐♠❛❧ ❞✐❣✐ts ✇❛s ❞✉❡ t♦ t❤❡ ●❡r♠❛♥ ♠❛t❤❡♠❛t✐❝✐❛♥▲✉❞♦❧♣❤ ✈❛♥ ❈❡✉❧❡♥ ✭✶✺✹✵✲✶✻✶✵✮✳ ❆♥♦t❤❡r ❊✉r♦♣❡❛♥ ❝♦♥tr✐❜✉t✐♦♥ t♦ t❤❡♣r♦❜❧❡♠ ✐s t❤❡ ❢♦r♠✉❧❛

2

π=

√2

2·√

2 +√2

2·

√

2 +√

2 +√2

2· · · · ✭✶✳✶✵✮

❢♦✉♥❞ ❜② ❋r❛♥❝♦✐s ❱✐ét❡ ✐♥ ✶✺✾✸✳ ❋♦r♠✉❧❛ ✶✳✶✵ ✇✐❧❧ ❜❡ ❞❡r✐✈❡❞ ✐♥ s❡❝t✐♦♥ ✶✵✳✸❜② ✉s✐♥❣ ✐♥s❝r✐❜❡❞ r❡❣✉❧❛r ♥✲❣♦♥s ✐♥ t❤❡ ✉♥✐t ❝✐r❝❧❡✳

✶✳✹ ❊①❡r❝✐s❡s

❊①❝❡r❝✐s❡ ✶✳✹✳✶ ❈♦♠♣✉t❡ t❤❡ ♥✉♠❜❡r ♦❢ st❡♣s ❢♦r t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢√2

✇✐t❤ ❡rr♦r ❧❡ss t❤❛♥ 10−10✳

❙♦❧✉t✐♦♥✳ ❲❡ ❤❛✈❡ t♦ s♦❧✈❡ t❤❡ ✐♥❡q✉❛❧✐t②

1

2n<

1

1010❢♦r t❤❡ ✉♥❦♥♦✇♥ ♥❛t✉r❛❧ ♥✉♠❜❡r n.

❊q✉✐✈❛❧❡♥t❧②✿ 1010 < 2n✳ ❚♦ s♦❧✈❡ t❤✐s ✐♥❡q✉❛❧✐t② ✇❡ ✉s❡ t❤❡ s♦✲❝❛❧❧❡❞ ❧♦❣❛✲r✐t❤♠ t♦ ❤❛✈❡ t❤❛t 10 log2 10 < n✳ ❙✐♥❝❡ 10 < 24 ✐t ❢♦❧❧♦✇s t❤❛t

10 log2 10 < 10 log2 24 = 40

st❡♣s ❛r❡ ❡♥♦✉❣❤ t♦ ❛♣♣r♦①✐♠❛t❡√2 ✇✐t❤ ❡rr♦r ❧❡ss t❤❛t 10−10 ✇❤✐❝❤ ✐s ❥✉st

t❤❡ ♠❡❛s✉r❡ ♦❢ t❤❡ ✉♥✐t ❝♦♥✈❡rs✐♦♥ ❜❡t✇❡❡♥ ♠❡t❡r ❛♥❞ ❆♥❣str♦♠ r❡❧❛t❡❞ t♦❛t♦♠✐❝✲s❝❛❧❡ str✉❝t✉r❡s✳

❊①❝❡r❝✐s❡ ✶✳✹✳✷ ❋✐♥❞ ❛ s❡q✉❡♥❝❡ ♦❢ r❛t✐♦♥❛❧ ♥✉♠❜❡rs t♦ ❛♣♣r♦①✐♠❛t❡√5✳

❙♦❧✉t✐♦♥✳ ❯s✐♥❣ t❤❡ ❡st✐♠❛t✐♦♥s

1 <√5 < 3

✇❡ ❤❛✈❡

q1 :=1 + 3

2= 2

❛s t❤❡ ✜rst ♠❡♠❜❡r ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♥❣ s❡q✉❡♥❝❡✳ ❙✐♥❝❡

2 <√5 < 3

✶✳✹✳ ❊❳❊❘❈■❙❊❙ ✷✼

✐t ❢♦❧❧♦✇s t❤❛t

q2 :=2 + 3

2=

5

2.

❘❡♣❡❛t✐♥❣ t❤❡ ❜❛s✐❝ st❡♣s ♦❢ t❤❡ ❞✐❝t✐♦♥❛r② ♠❡t❤♦❞ ✇❡ ❤❛✈❡

2 <√5 <

5

2⇒ q3 :=

2 + 5

2

2=

9

4,

2 <√5 <

9

4⇒ q4 :=

2 + 9

4

2=

17

8❛♥❞

17

8<

√5 <

9

4⇒ q5 :=

17

8+ 9

4

2=

35

16,

35

16<

√5 <

9

4⇒ q6 =

35

16+ 9

4

2=

71

32❛♥❞ s♦ ♦♥✳

❊①❝❡r❝✐s❡ ✶✳✹✳✸ ❈♦♥s✐❞❡r t❤❡ ✐t❡r❛t✐✈❡ s❡q✉❡♥❝❡

qn+1 =√

2 + qn,

✐✳❡✳

q1 =√2, q2 =

√

2 +√2, q3 =

√

2 +

√

2 +√2, . . .

Pr♦✈❡ t❤❛tqn ≤ 2

❢♦r ❛♥② ❡❧❡♠❡♥t ♦❢ t❤❡ s❡q✉❡♥❝❡✳

❙♦❧✉t✐♦♥✳ ■t ✐s ❝❧❡❛r t❤❛t t❤❡ ✐♥❡q✉❛❧✐t② ✐s tr✉❡ ❢♦r ♥❂✶✳ ❯s✐♥❣ ❛ s✐♠♣❧❡✐♥❞✉❝t✐♦♥

q2n+1 = 2 + qn ≤ 2 + 2 = 4.

❊①❝❡r❝✐s❡ ✶✳✹✳✹ ❋✐♥❞ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r s♦❧✉t✐♦♥s ❢♦r t❤❡ ❡q✉❛t✐♦♥

m2 −m− n = 0.

❙♦❧✉t✐♦♥✳ ❯s✐♥❣ t❤❡ ❢♦r♠✉❧❛ ❢♦r ❝♦♠♣✉t✐♥❣ t❤❡ r♦♦ts ♦❢ ❛ q✉❛❞r❛t✐❝ ❡q✉❛t✐♦♥

m12 =1±

√1 + 4n

2.

❚❤❡r❡❢♦r❡ ✶✰✹♥ ♠✉st ❜❡ ❛♥ ♦❞❞ sq✉❛r❡ ♥✉♠❜❡r✿

1 + 4n = (2k + 1)2,

✷✽ ❈❍❆P❚❊❘ ✶✳ ●❊◆❊❘❆▲ ❈❖▼P❯❚❆❚■❖◆❆▲ ❙❑■▲▲❙

1 + 4n = 4k2 + 4k + 1

✇❤✐❝❤ ♠❡❛♥s t❤❛t ♥ ♠✉st ❜❡ ♦❢ t❤❡ ❢♦r♠ ♥❂❦✭❦✰✶✮✱ ✇❤❡r❡ ❦ ✐s ❛♥ ❛r❜✐tr❛r②♣♦s✐t✐✈❡ ✐♥t❡❣❡r✳ ■♥ t❤✐s ❝❛s❡ t❤❡ ♣♦s✐t✐✈❡ r♦♦t ♦❢ t❤❡ ❡q✉❛t✐♦♥ ✐s ♠❂❦✰✶✳ ❋♦r❡①❛♠♣❧❡ ✐❢ ❦❂✶ t❤❡♥ ♥❂✷ ❛♥❞ ♠❂✷✳ ❋✉rt❤❡r ♣♦ss✐❜❧❡ s♦❧✉t✐♦♥s ❛r❡ ♥❂✶✷ ❛♥❞♠❂✹ ♦r ♥❂✷✵ ❛♥❞ ♠❂✺ ✉♥❞❡r t❤❡ ❝❤♦✐❝❡s ♦❢ ❦❂✸ ♦r ❦❂✹✳

❦ ♥❂❦✭❦✰✶✮ ♠❂❦✰✶✶ ✷ ✷✷ ✻ ✸✸ ✶✷ ✹✹ ✷✵ ✺✺ ✸✵ ✻

❊①❝❡r❝✐s❡ ✶✳✹✳✺ ❈♦♥s✐❞❡r t❤❡ ✐t❡r❛t✐✈❡ s❡q✉❡♥❝❡

qn+1 =√

12 + qn,

✐✳❡✳

q1 =√12, q2 =

√

12 +√12, q3 =

√

12 +

√

12 +√12, . . .

Pr♦✈❡ t❤❛tqn ≤ 4

❢♦r ❛♥② ❡❧❡♠❡♥t ♦❢ t❤❡ s❡q✉❡♥❝❡✳

❙♦❧✉t✐♦♥✳ ■t ✐s ❝❧❡❛r t❤❛t t❤❡ ✐♥❡q✉❛❧✐t② ✐s tr✉❡ ❢♦r ♥❂✶✳ ❯s✐♥❣ ❛ s✐♠♣❧❡✐♥❞✉❝t✐♦♥

q2n+1 = 12 + qn ≤ 12 + 4 = 16.

❊①❝❡r❝✐s❡ ✶✳✹✳✻ ❈♦♥s✐❞❡r t❤❡ ✐t❡r❛t✐✈❡ s❡q✉❡♥❝❡

qn+1 =√

20 + qn,

✐✳❡✳

q1 =√20, q2 =

√

20 +√20, q3 =

√

20 +

√

20 +√20, . . .

Pr♦✈❡ t❤❛tqn ≤ 5

❢♦r ❛♥② ❡❧❡♠❡♥t ♦❢ t❤❡ s❡q✉❡♥❝❡✳

❙♦❧✉t✐♦♥✳ ■t ✐s ❝❧❡❛r t❤❛t t❤❡ ✐♥❡q✉❛❧✐t② ✐s tr✉❡ ❢♦r ♥❂✶✳ ❯s✐♥❣ ❛ s✐♠♣❧❡✐♥❞✉❝t✐♦♥

q2n+1 = 20 + qn ≤ 20 + 5 = 25.

✶✳✹✳ ❊❳❊❘❈■❙❊❙ ✷✾

❘❡♠❛r❦ ❚❤❡ ✉♣♣❡r ❜♦✉♥❞s ✐♥ t❤❡ ♣r❡✈✐♦✉s ❡①❡r❝✐s❡s ♣r♦✈✐❞❡ t❤❛t t❤❡ s❡✲q✉❡♥❝❡s ❤❛✈❡ ✜♥✐t❡ ❧✐♠✐ts✳ ■♥ ❝❛s❡ ♦❢ t❤❡ s❡q✉❡♥❝❡

qn+1 =√

12 + qn

✇❡ ❤❛✈❡ t❤❛t t❤❡ ❧✐♠✐t ♠✉st s❛t✐s❢② t❤❡ ❡q✉❛t✐♦♥

q∗ =√

12 + q∗

❛♥❞✱ ❝♦♥s❡q✉❡♥t❧②✱ ✐t ✐s ❥✉st ✹✳

❊①❝❡r❝✐s❡ ✶✳✹✳✼ ❋✐♥❞ t❤❡ ❧✐♠✐t ♦❢ t❤❡ s❡q✉❡♥❝❡

qn+1 =√

2 + qn.

❙♦❧✉t✐♦♥✳ ❆s ✇❡ ❤❛✈❡ s❡❡♥ ❛❜♦✈❡ t❤❡ s❡q✉❡♥❝❡ ✐s ❜♦✉♥❞❡❞ ❜② ✷ ❢r♦♠ ❛❜♦✈❡✳❚❤✐s ♠❡❛♥s t❤❛t ✇❡ ❤❛✈❡ ❛ ✜♥✐t❡ ❧✐♠✐t s❛t✐s❢②✐♥❣ t❤❡ ❡q✉❛t✐♦♥

q∗ =√

2 + q∗.

❚❤❡r❡❢♦r❡0 = q2∗ − q∗ − 2

✇❤✐❝❤ ♠❡❛♥s t❤❛t q∗ = 2 ♦r ✲ ✶ ❜✉t t❤❡ ♥❡❣❛t✐✈❡ ✈❛❧✉❡ ❝❛♥ ❜❡ ♦❜✈✐♦✉s❧②♦♠✐tt❡❞✳

❊①❝❡r❝✐s❡ ✶✳✹✳✽ ❋✐♥❞ t❤❡ ❧✐♠✐t ♦❢ t❤❡ s❡q✉❡♥❝❡

qn+1 =√

20 + qn.

❙♦❧✉t✐♦♥✳ ❆s ✇❡ ❤❛✈❡ s❡❡♥ ❛❜♦✈❡ t❤❡ s❡q✉❡♥❝❡ ✐s ❜♦✉♥❞❡❞ ❜② ✺ ❢r♦♠ ❛❜♦✈❡✳❚❤✐s ♠❡❛♥s t❤❛t ✇❡ ❤❛✈❡ ❛ ✜♥✐t❡ ❧✐♠✐t s❛t✐s❢②✐♥❣ t❤❡ ❡q✉❛t✐♦♥

q∗ =√

20 + q∗.

❚❤❡r❡❢♦r❡0 = q2∗ − q∗ − 20

✇❤✐❝❤ ♠❡❛♥s t❤❛t q∗ = 5 ♦r ✲ ✹ ❜✉t t❤❡ ♥❡❣❛t✐✈❡ ✈❛❧✉❡ ❝❛♥ ❜❡ ♦❜✈✐♦✉s❧②♦♠✐tt❡❞✳

❊①❝❡r❝✐s❡ ✶✳✹✳✾ Pr♦✈❡ t❤❛t

a2 − 1 = (a− 1)(a+ 1).

✸✵ ❈❍❆P❚❊❘ ✶✳ ●❊◆❊❘❆▲ ❈❖▼P❯❚❆❚■❖◆❆▲ ❙❑■▲▲❙

❙♦❧✉t✐♦♥✳ ■t ❝❛♥ ❜❡ ❡❛s✐❧② ❞❡r✐✈❡❞ ❜② ❞✐r❡❝t ❝❛❧❝✉❧❛t✐♦♥✿

(a− 1)(a+ 1) = a2 + a− a− 1 = a2 − 1.

❊①❝❡r❝✐s❡ ✶✳✹✳✶✵ Pr♦✈❡ t❤❛t

a3 − 1 = (a− 1)(a2 + a+ 1).

❙♦❧✉t✐♦♥✳ ■t ❝❛♥ ❜❡ ❡❛s✐❧② ❞❡r✐✈❡❞ ❜② ❞✐r❡❝t ❝❛❧❝✉❧❛t✐♦♥✿

(a− 1)(a2 + a+ 1) = a3 + a2 + a− a2 − a− 1 = a3 − 1.

❚❤❡ ❢♦r♠✉❧❛s ✐♥✈♦❧✈✐♥❣ ❡①♣❧✐❝✐t❡ ♣♦✇❡rs ❝❛♥ ❜❡ ❣✐✈❡♥ ❜② t❤❡ ❤❡❧♣ ♦❢ ❞✐r❡❝t❝❛❧❝✉❧❛t✐♦♥s✳

❊①❝❡r❝✐s❡ ✶✳✹✳✶✶ Pr♦✈❡ t❤❛t ❢♦r ❛♥② ♥❛t✉r❛❧ ♣♦✇❡r

an − 1 = (a− 1)(an−1 + an−2 + . . .+ a+ 1).

❙♦❧✉t✐♦♥✳ ▲❡tsn−1 = 1 + a+ . . .+ an−1

❜❡ t❤❡ ♣❛rt✐❛❧ s✉♠ ♦❢ t❤❡ ♣♦✇❡rs✳ ❚❤❡♥

(a− 1)sn−1 = asn−1 − sn−1 = a+ a2 + . . .+ an − (1 + a+ . . .+ an−1) =

an − 1

❛s ✇❛s t♦ ❜❡ ♣r♦✈❡❞✳

❘❡♠❛r❦ ❯s❡ t❤❡ ♣r♦❝❡❞✉r❡ ♦❢ t❤❡ ✐♥❞✉❝t✐♦♥ t♦ ♣r♦✈❡ t❤❡ st❛t❡♠❡♥t ✐♥ ❊①✲❡r❝✐s❡ ✶✳✹✳✶✶✳

❙♦❧✉t✐♦♥✳

an+1 − 1 = an+1 − an + an − 1 = an(a− 1) + t❤❡ ✐♥❞✉❝t✐✈❡ ❤②♣♦t❤❡s✐s...

❊①❝❡r❝✐s❡ ✶✳✹✳✶✷ ❈❛❧❝✉❧❛t❡ t❤❡ s✉♠ ♦❢ t❤❡ s❡r✐❡s

1 +1

2+

1

22+

1

23+ . . .

❍✐♥t✳ ❯s✐♥❣ t❤❡ ♣r❡✈✐♦✉s r❡s✉❧t ✇✐t❤ a = 1/2 ✇❡ ❤❛✈❡ t❤❛t

1 +1

2+

1

22+

1

23+ . . .+

1

2n−1=

1

2n− 1

1

2− 1

→ −11

2− 1

= 2.

❘❡♠❛r❦ ❲❡ ❝❛♥ ✐♠❛❣❡ t❤❡ s✉♠ ♦❢ t❤❡ ❣❡♦♠❡tr✐❝ s❡r✐❡s

1 +1

2+

1

22+

1

23+ . . .

❛s t❛❦✐♥❣ ❛ ✷ ✉♥✐ts ❧♦♥❣ ✇❛❧❦ ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t ❡❛❝❤ s✉❜ ✲ ✇❛❧❦ t❛❦❡s t❤❡❤❛❧❢ ♦❢ t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ st❛r✐♥❣ ♣♦✐♥t t♦ t❤❡ ❡♥❞✳

✶✳✺✳ ❋❯◆❈❚■❖◆❙ ✸✶

❋✐❣✉r❡ ✶✳✺✿ ❊①♣♦♥❡♥t✐❛❧❧② ❞❡❝r❡❛s✐♥❣ t❡♥❞❡♥❝②✳

✶✳✺ ❋✉♥❝t✐♦♥s

❚❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ sq✉❛r❡ r♦♦t ✷ ❝❛♥ ❜❡ ✐♥t❡r♣r❡t❡❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✇❛②✳

❙t❡♣ ❱❛❧✉❡ ❇♦✉♥❞ ❢♦r t❤❡ ❡rr♦r✶st q1 = 3/2 ✶✴✷✷♥❞ q2 = 5/4 ✶✴22

✸r❞ q3 = 8/11 ✶✴23

✳✳✳ ✳✳✳♥ ✲ t❤ qn 1/2n

❇❡s✐❞❡s t❤❡ t❛❜✉❧❛r ❢♦r♠ ❣r❛♣❤✐❝❛❧ r❡♣r❡s❡♥t❛t✐♦♥ ✐s ✇✐❞❡❧② ✉s❡❞✳ ❆❝t✉❛❧❧②t❤✐s ✐s ❛ ❞✐r❡❝t ♠❡t❤♦❞ t♦ r❡❛❧✐③❡ r❡❧❛t✐♦♥s❤✐♣s ❛♥❞ t❡♥❞❡♥❝✐❡s ❛♠♦♥❣ ❞❛t❛✐t❡♠s ❛t ❛ ❣❧❛♥❝❡✳

✶✳✺✳✶ ❊①♣♦♥❡♥t✐❛❧s

❊①♣♦♥❡♥t✐❛❧s ❛r❡ t②♣✐❝❛❧ ✐♥ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧✐♥❣ ♦❢ ❣r♦✇✐♥❣ ✇✐t❤♦✉t ❝♦♥✲str❛✐♥ts ✭s❡❡ ❡❣✳ ❝❡❧❧ ❞✐✈✐s✐♦♥✱ ❢❛♠✐❧② tr❡❡✮✳ ❲❡ ❛❧s♦ ❦♥♦✇ t❤❛t ❡❛❝❤ r❛❞✐♦❛❝✲t✐✈❡ ✐s♦t♦♣❡ ❤❛s ✐ts ♦✇♥ ❝❤❛r❛❝t❡r✐st✐❝ ❞❡❝❛② ♣❛tt❡r♥✳ ■ts r❛t❡ ✐s ♠❡❛s✉r❡❞✐♥ ❤❛❧❢ ✲ ❧✐❢❡✳ ❚❤❡ ❤❛❧❢ ✲ ❧✐❢❡ r❡❢❡rs t♦ t❤❡ t✐♠❡ ✐t t❛❦❡s ❢♦r ♦♥❡ ✲ ❤❛❧❢ ♦❢t❤❡ ❛t♦♠s ♦❢ ❛ r❛❞✐♦❛❝t✐✈❡ ♠❛t❡r✐❛❧ t♦ ❞✐s✐♥t❡❣r❛t❡✳ ❍❛❧❢ ✲ ❧✐✈❡s ❢♦r ❞✐✛❡r❡♥tr❛❞✐♦✐s♦t♦♣❡s ❝❛♥ r❛♥❣❡ ❢r♦♠ ❛ ❢❡✇ ♠✐❝r♦s❡❝♦♥❞ t♦ ❜✐❧❧✐♦♥s ♦❢ ②❡❛rs✳

✸✷ ❈❍❆P❚❊❘ ✶✳ ●❊◆❊❘❆▲ ❈❖▼P❯❚❆❚■❖◆❆▲ ❙❑■▲▲❙

❘❛❞✐♦✐s♦t♦♣❡ ❍❛❧❢ ✲ ❧✐❢❡P♦❧♦♥✐✉♠✲✷✶✺ ✵✳✵✵✶✽ s❡❝♦♥❞s❇✐s♠✉t✲✷✶✷ ✻✵✳✺ s❡❝♦♥❞s❇❛r✐✉♠✲✶✸✾ ✽✻ ♠✐♥✉t❡s❙♦❞✐✉♠✲✷✹ ✶✺ ❤♦✉rs❈♦❜❛❧t✲✻✵ ✺✳✷✻ ②❡❛rs❘❛❞✐✉♠✲✷✷✻ ✶✻✵✵ ②❡❛rs❯r❛♥✐✉♠✲✷✸✽ ✹✳✺ ❜✐❧❧✐♦♥ ②❡❛rs

✶✳✺✳✷ ❚r✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s

❆♥♦t❤❡r ✐♠♣♦rt❛♥t t②♣❡ ♦❢ ❢✉♥❝t✐♦♥s ❛r❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s❀ s❡❡ s❡❝t✐♦♥✹✳✺✳

✶✳✺✳✸ P♦❧②♥♦♠✐❛❧s

❋✐♥❛❧❧② ✇❡ ♠❡♥t✐♦♥ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ♦❢ t❤❡ ❢♦r♠

f(x) = anxn + an−1x

n−1 + . . .+ a1x+ a0.

❚❤❡ ♠♦st ✐♠♣♦rt❛♥t s♣❡❝✐❛❧ ❝❛s❡s ❛r❡ ♥❂✶ ✭❧✐♥❡s✮ ❛♥❞ ♥❂✷ ✭♣❛r❛❜♦❧❛s✮✳ P♦❧②✲♥♦♠✐❛❧s ❜❡❤❛✈❡ ❧✐❦❡ ♥✉♠❜❡rs ❢r♦♠ s♦♠❡ ♣♦✐♥ts ♦❢ ✈✐❡✇✳ ❲❡ ❝❛♥ ❛❞❞ ♦r ♠✉❧✲t✐♣❧② t❤❡♠ ❛♥❞ ✇❡ ❝❛♥ ❞✐✈✐❞❡ t✇♦ ♣♦❧②♥♦♠✐❛❧s ✇✐t❤ ❡❛❝❤ ♦t❤❡r t♦♦✳

2x3 + x2 − 1 ✿ x+ 1 = 2x2

− (2x3 + 2x2)

−x2 − 1 ✿ x+ 1 ❂−x− (−x2 − x)

x− 1 ✿ x+ 1 = 1− (x+ 1)

−2

❚❤❡r❡❢♦r❡2x3 + x2 − 1 = (2x2 − x+ 1)(x+ 1)− 2.

❆♥ ✐♠♣♦rt❛♥t ❡①❛♠♣❧❡ ♦♥ ❛ ♣♦❧②♥♦♠✐❛❧ t❡♥❞❡♥❝② ✐s t❤❡ ❦✐♥❡♠❛t✐❝ ❧❛✇

❢♦r t❤❡ ❞✐st❛♥❝❡ tr❛✈❡❧❧❡❞ ❞✉r✐♥❣ ❛ ✉♥✐❢♦r♠ ❛❝❝❡❧❡r❛t✐♦♥ st❛rt✐♥❣ ❢r♦♠ r❡st✳■t ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ sq✉❛r❡ ♦❢ t❤❡ ❡❧❧❛♣s❡❞ t✐♠❡✳ ❚❤✐s ✐s t❤❡ s✐t✉❛t✐♦♥✐♥ ❝❛s❡ ♦❢ ❢❛❧❧✐♥❣ ❜♦❞✐❡s ✐♥✈❡st✐❣❛t❡❞ ❜② ●❛❧✐❧❡♦ ●❛❧✐❧❡✐✳ ■❢ ✇❡ ❛r❡ ✐♥t❡r❡st❡❞

✶✳✻✳ ❊❳❊❘❈■❙❊❙ ✸✸

✐♥ t❤❡ ❞✐st❛♥❝❡ tr❛✈❡❧❧❡❞ ❜② ❛ ❢❛❧❧✐♥❣ ❜♦❞② ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ tr❛✈❡❧❧✐♥❣t✐♠❡ ✐t ✐s r❡❧❛t✐✈❡❧② ❤❛r❞ t♦ ❝r❡❛t❡ ❛♥ ❛♣♣r♦♣r✐❛t❡ ❡①♣❡r✐♠❡♥t❛❧ ❡♥✈✐r♦♥♠❡♥t❢♦r ♠❡❛s✉r✐♥❣✳ ■t ✐s ♠♦r❡ r❡❛s♦♥❛❜❧❡ t♦ ♠❡❛s✉r❡ t❤❡ tr❛✈❡❧❧✐♥❣ t✐♠❡ ❛s t❤❡❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❞✐st❛♥❝❡✳ ■♥ ♦t❤❡r ✇♦r❞s ✇❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ t❤❡ ✐♥✈❡rs❡ r❡❧❛✲t✐♦♥s❤✐♣ ✭✐♥✈❡rs❡ ❢✉♥❝t✐♦♥✮✳ ❚♦ ❝r❡❛t❡ ❛ ❝♦♠❢♦rt❛❜❧❡ ❡①♣❡r✐♠❡♥t❛❧ s✐t✉❛t✐♦♥✇❡ ❝❛♥ ✉s❡ ❛ s❧♦♣❡ t♦ ❡♥s✉r❡ ❛ tr❛✈❡❧ ❞✉r✐♥❣ ❛ ✉♥✐❢♦r♠ ❛❝❝❡❧❡r❛t✐♦♥ st❛rt✐♥❣❢r♦♠ r❡st✳ ❆ s✐♠♣❧❡ s❝❛❧❡ ❝❛♥ ❜❡ ❣✐✈❡♥ ❜② ✉s✐♥❣ t❤❡ ♠✐❞✲♣♦✐♥t t❡❝❤♥✐❝ ❛❧♦♥❣t❤❡ s❧♦♣❡✳ ❚❤❡♦r❡t✐❝❛❧❧② ✇❡ ❤❛✈❡ t❤❡ ❢♦r♠✉❧❛

f(s) =

√

2s

a

t♦ ❣✐✈❡ t❤❡ tr❛✈❡❧❧✐♥❣ t✐♠❡ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❞✐st❛♥❝❡ s ❛❧♦♥❣ t❤❡ s❧♦♣❡❀ t❤❡❝♦♥st❛♥t

a = g sinα

✐s r❡❧❛t❡❞ t♦ t❤❡ ❛♥❣❧❡ ♦❢ t❤❡ s❧♦♣❡ ❛♥❞ t❤❡ ❣r❛✈✐t❛t✐♦♥❛❧ ❛❝❝❡❧❡r❛t✐♦♥ ❣✳ ❚♦r❡t✉r♥ t♦ t❤❡ ♦r✐❣✐♥❛❧ ♣r♦❜❧❡♠ ✇❡ ♥❡❡❞ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❢✳ ❋♦r♠❛❧❧②s♣❡❛❦✐♥❣ ✇❡ ✇❛♥t t♦ ❡①♣r❡ss s ✐♥ t❡r♠s ♦❢ t❂❢✭s✮✿

t =

√

2s

a⇒ a

2t2 = s

❛♥❞✱ ❝♦♥s❡q✉❡♥t❧②✱ t❤❡ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥ ✐s ✇♦r❦✐♥❣ ❛s

f−1(t) =a

2t2

♦♥ t❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ ♥♦♥✲♥❡❣❛t✐✈❡ r❡❛❧ ♥✉♠❜❡rs✳ ●❡♦♠❡tr✐❝❛❧❧② ✇❡ ❝❤❛♥❣❡t❤❡ r♦❧❡ ♦❢ t❤❡ ❝♦♦r❞✐♥❛t❡s ① ❛♥❞ ② ✐♥ t❤❡ ❝♦♦r❞✐♥❛t❡ ♣❧❛♥❡✳ ❚❤❡r❡❢♦r❡ t❤❡❣r❛♣❤s ♦❢ ❛ ❢✉♥❝t✐♦♥ ❛♥❞ ✐ts ✐♥✈❡rs❡ ✐s r❡❧❛t❡❞ ❜② t❤❡ r❡✢❡❝t✐♦♥ ❛❜♦✉t t❤❡ ❧✐♥❡②❂① ❛s ✇❡ ❝❛♥ s❡❡ ✐♥ t❤❡ ♥❡①t ✜❣✉r❡ ❢♦r t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❛♥❞ t❤❡ ❧♦❣❛r✐t❤♠✐❝❢✉♥❝t✐♦♥s✳

✶✳✻ ❊①❡r❝✐s❡s

❊①❝❡r❝✐s❡ ✶✳✻✳✶ ❙✉♣♣♦s❡ t❤❛t ②♦✉ ❤❛✈❡ ✶✵ ❣r❛♠s ♦❢ ❇❛r✐✉♠ ✲ ✶✸✾✳ ❆❢t❡r✽✻ ♠✐♥✉t❡s✱ ❤❛❧❢ ♦❢ t❤❡ ❛t♦♠s ✐♥ t❤❡ s❛♠♣❧❡ ✇♦✉❧❞ ❤❛✈❡ ❞❡❝❛②❡❞ ✐♥t♦ ❛♥♦t❤❡r❡❧❡♠❡♥t ❝❛❧❧❡❞ ▲❛♥t❤❛♥✉♠ ✲ ✶✸✾✳ ❆❢t❡r ♦♥❡ ❤❛❧❢ ✲ ❧✐❢❡ ②♦✉ ✇♦✉❧❞ ❤❛✈❡ ✺ ❣r❛♠s❇❛r✐✉♠ ✲ ✶✸✾ ❛♥❞ ✺ ❣r❛♠s ▲❛♥t❤❛♥✉♠ ✲ ✶✸✾✳ ❆❢t❡r ❛♥♦t❤❡r ✽✻ ♠✐♥✉t❡s✱ ❤❛❧❢♦❢ t❤❡ ✺ ❣r❛♠s ❇❛r✐✉♠ ✲ ✶✸✾ ✇♦✉❧❞ ❞❡❝❛② ✐♥t♦ ▲❛♥t❤❛♥✉♠ ✲ ✶✸✾ ❛❣❛✐♥❀ ②♦✉✇♦✉❧❞ ♥♦✇ ❤❛✈❡ ✷✳✺ ❣r❛♠s ♦❢ ❇❛r✐✉♠ ✲ ✶✸✾ ❛♥❞ ✼✳✺ ❣r❛♠s ▲❛♥t❤❛♥✉♠ ✲ ✶✸✾✳❍♦✇ ♠❛♥② t✐♠❡ ❞♦❡s ✐t t❛❦❡ t♦ ❜❡ ❇❛r✐✉♠ ✲ ✶✸✾ ❧❡ss t❤❛♥ ✶ ❣r❛♠❄

✸✹ ❈❍❆P❚❊❘ ✶✳ ●❊◆❊❘❆▲ ❈❖▼P❯❚❆❚■❖◆❆▲ ❙❑■▲▲❙

❋✐❣✉r❡ ✶✳✻✿ ❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ❛♥❞ ✐ts ✐♥✈❡rs❡✳

❚✐♠❡ ✭♠✐♥✉t❡s✮ ❛♠♦✉♥t ♦❢ ❇❛r✐✉♠ ✲ ✶✸✾ ✭❣r❛♠✮✵ ✶✵✽✻ ✺

✷ × ✽✻ ✷✳✺✸ × ✽✻ ✶✳✷✺✹ × ✽✻ ✵✳✻✷✺

✳✳✳ ✳✳✳♥ × ✽✻ 10/2n

❙♦❧✉t✐♦♥✳ ❲❡ ❤❛✈❡ t♦ s♦❧✈❡ t❤❡ ✐♥❡q✉❛❧✐t②

10

2n< 1 ⇒ 10 < 2n.

❚❤❡r❡❢♦r❡ 4 · 86 ♠✐♥✉t❡s ✐s ❡♥♦✉❣❤ t♦ ❜❡ ❇❛r✐✉♠ ✲ ✶✸✾ ❧❡ss t❤❛♥ ✶ ❣r❛♠✳

❊①❝❡r❝✐s❡ ✶✳✻✳✷ ❙❦❡t❝❤ t❤❡ ❢✉♥❝t✐♦♥s f(x) = 2x ❛♥❞ g(x) = log2 x ✐♥ ❛❝♦♠♠♦♥ ❈❛rt❡s✐❛♥ ❝♦♦r❞✐♥❛t❡ s②st❡♠✳

❊①❝❡r❝✐s❡ ✶✳✻✳✸ Pr♦✈❡ t❤❛t log2 3 ✐s ✐rr❛t✐♦♥❛❧✳

✶✳✻✳ ❊❳❊❘❈■❙❊❙ ✸✺

❙♦❧✉t✐♦♥✳ ❙✉♣♣♦s❡✱ ✐♥ ❝♦♥tr❛r② t❤❛t

log2 3 =n

m,

✇❤❡r❡ ♥ ❛♥❞ ♠6= 0 ❛r❡ ✐♥t❡❣❡rs✳ ❯s✐♥❣ t❤❛t log2 3m = m log2 3 ✇❡ ❤❛✈❡

log2 3m = n.

❇② t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❧♦❣❛r✐t❤♠ t❤✐s ♠❡❛♥s t❤❛t 2n = 3m ✇❤✐❝❤ ✐s ♦❜✈✐♦✉s❧②✐♠♣♦ss✐❜❧❡✳

❊①❝❡r❝✐s❡ ✶✳✻✳✹ ❚r❛♥s❢❡r t❤❡ ❡①♣r❡ss✐♦♥ f(x) = 3x2−5x+3 t♦ t❤❡ ❝❛♥♦♥✐❝❛❧❢♦r♠

f(x) = a(x− x0)2 + y0

❛♥❞ ❝♦♠♣✉t❡ t❤❡ ♠✐♥✐♠✉♠ ✈❛❧✉❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳

❙♦❧✉t✐♦♥✳ ■t ❝❛♥ ❜❡ ❡❛s✐❧② s❡❡♥ t❤❛t

f(x) = 3(

x2 − (5/3)x+ 1)

= 3(

(x− (5/6))2 − (25/36) + 1)

=

= 3 (x− (5/6))2 + (11/12),

✐✳❡✳ t❤❡ ♠✐♥✐♠✉♠ ✈❛❧✉❡ ✐s ❥✉st y0 = 11/12 ❛tt❛✐♥❡❞ ❛t x0 = 5/6✳

❊①❝❡r❝✐s❡ ✶✳✻✳✺ Pr♦✈❡ t❤❡ ❢♦r♠✉❧❛

x12 =−b±

√b2 − 4ac

2a

❢♦r t❤❡ r♦♦ts ♦❢ t❤❡ ❡q✉❛t✐♦♥

ax2 + bx+ c = 0

❜② ✉s✐♥❣ t❤❡ ❝❛♥♦♥✐❝❛❧ ❢♦r♠ ♦❢ ❛ q✉❛❞r❛t✐❝ ❢✉♥❝t✐♦♥✳

❍✐♥t✳ ❈♦♥s✐❞❡r t❤❡ ❢✉♥❝t✐♦♥

f(x) = ax2 + bx+ c.

■ts ❝❛♥♦♥✐❝❛❧ ❢♦r♠ ✐s

f(x) = a

(

x+b

2a

)2

− b2

4a+ c

✸✻ ❈❍❆P❚❊❘ ✶✳ ●❊◆❊❘❆▲ ❈❖▼P❯❚❆❚■❖◆❆▲ ❙❑■▲▲❙

✇❤✐❝❤ ✐♠♣❧✐❡s ❜② t❛❦✐♥❣ t❤❡ ❡q✉❛t✐♦♥ ❢✭①✮❂✵ t❤❛t(

x+b

2a

)2

=b2 − 4ac

4a2.

❚❤❡r❡❢♦r❡

x+b

2a= ±

√b2 − 4ac

2a♣r♦✈✐❞❡❞ t❤❛t t❤❡ ❞✐s❝r✐♠✐♥❛♥t D = b2 − 4ac ✐s ♥♦♥✲♥❡❣❛t✐✈❡✳

❊①❝❡r❝✐s❡ ✶✳✻✳✻ ❈♦♥❝❧✉❞❡ ❱✐ét❡✬s ❢♦r♠✉❧❛s

x1 + x2 = − b

a❛♥❞ x1 · x2 =

c

a.

❊①❝❡r❝✐s❡ ✶✳✻✳✼ ❋✐♥❞ t❤❡ ♠❛①✐♠✉♠ ❛♠♦✉♥t ♦❢ sq✉❛r❡ ❢♦♦t❛❣❡ ✇❡ ❝❛♥ ❡♥✲❝❧♦s❡ ✐♥ ❛ r❡❝t❛♥❣❧❡ ✉s✐♥❣ ❛ ❢❡♥❝❡ ✇✐t❤ ✶✷✽ ❢❡❡t✳

❙♦❧✉t✐♦♥✳ ▲❡t ① ❛♥❞ ② ❜❡ t❤❡ s✐❞❡s ♦❢ ❛ r❡❝t❛♥❣❧❡✳ ❚♦ ✜♥❞ t❤❡ ♠❛①✐♠✉♠ ♦❢t❤❡ ♣r♦❞✉❝t ①② s✉❜❥❡❝t t♦ t❤❡ ❡q✉❛❧✐t② ❝♦♥str❛✐♥ ✷✭①✰②✮❂✶✷✽ ❝♦♥s✐❞❡r t❤❡❢✉♥❝t✐♦♥

f(x, y) = xy.

❙✉❜st✐t✉t✐♥❣ ②❂✻✹ ✲ ① ✇❡ ❝❛♥ r❡❞✉❝❡ t❤❡ ♥✉♠❜❡r ♦❢ ✈❛r✐❛❜❧❡s✿

f(x) = x(64− x) = −x2 + 64x = −(x− 32)2 + 322.

❚❤❡ ♠❛①✐♠✉♠ ❛r❡❛ ✐s ❥✉st ✶✵✷✹ ❛tt❛✐♥❡❞ ❛t ①❂✸✷ ✇❤✐❝❤ ✐s ❥✉st t❤❡ ❝❛s❡ ♦❢ ❛sq✉❛r❡✳

❊①❝❡r❝✐s❡ ✶✳✻✳✽ ❚❤❡ ❢♦❧❧♦✇✐♥❣ t❛❜❧❡ s❤♦✇s t❤❡ ❛✈❡r❛❣❡ ❤✐❣❤s ♦❢ t❡♠♣❡r❛t✉r❡♠❡❛s✉r❡❞ ♦♥ ✶✺t❤ ♦❢ ❡❛❝❤ ♠♦♥t❤ ✐♥ ◆❡✇ ❨♦r❦ ❈✐t② ❬✸❪✳ ❯s✐♥❣ ❣r❛♣❤✐❝❛❧r❡♣r❡s❡♥t❛t✐♦♥ ✜♥❞ t❤❡ r✉❧❡ ♦❢ t❤❡ ❛✈❡r❛❣❡ ❤✐❣❤s✳ ❲❤❛t ❛❜♦✉t t❤❡ t❡♠♣❡r❛t✉r❡♦♥ ✸✵t❤ ♦❢ ❖❝t♦❜❡r❄

▼♦♥t❤ ❚❡♠♣❡r❛t✉r❡ ✭❋❛❤r❡♥❤❡✐t✮❋❡❜r✉❛r② ✹✵▼❛r❝❤ ✺✵❆♣r✐❧ ✻✷▼❛② ✼✷❏✉♥② ✽✶❏✉❧② ✽✺

❆✉❣✉st ✽✸❙❡♣t❡♠❜❡r ✼✽❖❝t♦❜❡r ✻✻◆♦✈❡♠❜❡r ✺✻❉❡❝❡♠❜❡r ✹✵

✶✳✻✳ ❊❳❊❘❈■❙❊❙ ✸✼

❋✐❣✉r❡ ✶✳✼✿ ❚❤❡ ❣r❛♣❤✐❝❛❧ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❤✐❣❤ t❡♠♣❡r❛t✉r❡s✳

❙♦❧✉t✐♦♥✳ ❈♦♥s✐❞❡r t❤❡ ♠♦♥t❤s ❛s ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡s ①❂✷✱ ✸✱ ✹✱ ✺✱ ✻✱ ✼✱ ✽✱✾✱ ✶✵✱ ✶✶✱ ✶✷✳ ❋♦r t❤❡ s❛❦❡ ♦❢ s✐♠♣❧✐❝✐t② ✇❡ ✐❧❧✉str❛t❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❤✐❣❤t❡♠♣❡r❛t✉r❡s ❛s ❚❂✹✱ ✺✱ ✻✳✷✱ ✳✳✳ ❛♥❞ s♦ ♦♥✳ ❆s ✐t ❝❛♥ ❜❡❡ s❡❡♥ t❤❡② ❢♦r♠ ❛♣❛r❛❜♦❧✐❝ ❛r❝ ✇✐t❤ ❝❛♥♦♥✐❝❛❧ ❢♦r♠

f(x) = a(x− 7)2 + 8.5.

❚♦ ❝♦♠♣✉t❡ t❤❡ ♣❛r❛♠❡t❡r ✑❛✑ ✇❡ ❝❛♥ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ s✉❜st✐t✉t✐♦♥s✿

4 = a(2− 7)2 + 8.5 ⇒ a = −0.18,

5 = a(3− 7)2 + 8.5 ⇒ a = −0.21,

6.2 = a(4− 7)2 + 8.5 ⇒ a = −0.25

❛♥❞ s♦ ♦♥✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ t❛❜❧❡ s❤♦✇s t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ t❤❡ ♣♦ss✐❜❧❡ ✈❛❧✉❡s ♦❢t❤❡ ♣❛r❛♠❡t❡r ✧❛✧✳

① ❢✭①✮❂❛✭①✲✼✮✰✽✳✺ ❛✷ ✭❋❡❜r✉❛r②✮ 4 = a(2− 7)2 + 8.5 ✲✵✳✶✽✸ ✭▼❛r❝❤✮ 5 = a(3− 7)2 + 8.5 ✲✵✳✷✶✹ ✭❆♣r✐❧✮ 6.2 = a(4− 7)2 + 8.5 ✲✵✳✷✺✺ ✭▼❛②✮ 7.2 = a(5− 7)2 + 8.5 ✲✵✳✸✷✻ ✭❏✉♥②✮ 8.1 = a(6− 7)2 + 8.5 ✲✵✳✹✼ ✭❏✉❧②✮ 8.5 = a(7− 7)2 + 8.5 ✲

✽ ✭❆✉❣✉st✮ 8.3 = a(8− 7)2 + 8.5 ✲✵✳✷✾ ✭❙❡♣t❡♠❜❡r✮ 7.8 = a(9− 7)2 + 8.5 ✲✵✳✶✼

❖❝t♦❜❡r◆♦✈❡♠❜❡r❉❡❝❡♠❜❡r

✸✽ ❈❍❆P❚❊❘ ✶✳ ●❊◆❊❘❆▲ ❈❖▼P❯❚❆❚■❖◆❆▲ ❙❑■▲▲❙

❊①❝❡r❝✐s❡ ✶✳✻✳✾ ❈❛❧❝✉❧❛t❡ t❤❡ ♠✐ss✐♥❣ ✈❛❧✉❡s ♦❢ t❤❡ ♣❛r❛♠❡t❡r✳

❙♦❧✉t✐♦♥✳ ❯s✐♥❣ t❤❡ ❡q✉❛t✐♦♥s

6.6 = a(10− 7)2 + 8.5,

5.6 = a(11− 7)2 + 8.5,

4 = a(12− 7)2 + 8.5

✇❡ ❤❛✈❡ t❤❡ ✈❛❧✉❡s ❛❂ ✲ ✵✳✷✶✱ ✲ ✵✳✶✽ ❛♥❞ ✲ ✵✳✶✽✳ ❚❤❡r❡❢♦r❡ t❤❡ ♣❛r❛♠❡t❡r ✑❛✑✐s ❛❜♦✉t ✲ ✵✳✷✳ ❆ r❡❛s♦♥❛❜❧❡ ♠♦❞❡❧ t♦ ❝♦♠♣✉t❡ t❤❡ ❛✈❡r❛❣❡ ❤✐❣❤ t❡♠♣❡r❛t✉r❡✐s

T (x)/10 = −0.2(x− 7)2 + 8.5.

✸✵✴❖❝t♦❜❡r ❝♦rr❡s♣♦♥❞s t❤❡ ✈❛❧✉❡ ①❂✶✵✳✺✳ ❚❤❡r❡❢♦r❡

T (10.5) = −2(10.5− 7)2 + 85 = 60.5

❋❛❤r❡♥❤❡✐t✳

❊①❝❡r❝✐s❡ ✶✳✻✳✶✵ ❋✐♥❞ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥

f(x) = 3x− 4.

❙♦❧✉t✐♦♥✳ ❊①♣r❡ss ① ✐♥ t❡r♠s ♦❢ ②❂❢✭①✮✿

y = 3x− 4 ⇒ x =y + 4

3=

1

3y +

4

3

✇❤✐❝❤ ♠❡❛♥s t❤❛t t❤❡ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥ ✐s ✇♦r❦✐♥❣ ❛s

f−1(y) =1

3y +

4

3.

❊①❝❡r❝✐s❡ ✶✳✻✳✶✶ ❋✐♥❞ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥

f(x) = x2.

❙♦❧✉t✐♦♥✳ ❚❤❡ ❢♦r♠❛❧ ♠❡t❤♦❞ ❣✐✈❡s t❤❛t

y = x2 ⇒ x =√y

❛♥❞✱ ❝♦♥s❡q✉❡♥t❧②✱ t❤❡ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥ ✐s ✇♦r❦✐♥❣ ❛s

f−1(y) =√y

♦♥ t❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ ♥♦♥✲♥❡❣❛t✐✈❡ r❡❛❧ ♥✉♠❜❡rs✳

✶✳✻✳ ❊❳❊❘❈■❙❊❙ ✸✾

❊①❝❡r❝✐s❡ ✶✳✻✳✶✷ ❋✐♥❞ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥

f(x) =3x− 4

x− 2.

❙♦❧✉t✐♦♥✳ ❚❤❡ ❢♦r♠❛❧ ♠❡t❤♦❞ ❣✐✈❡s t❤❛t

y =3x− 4

x− 2,

yx− 2y − 3x+ 4 = 0,

x(y − 3)− 2y + 4 = 0,

x =2y − 4

y − 3.

❚❤❡r❡❢♦r❡

f−1(y) =2y − 4

y − 3

❛♥❞ t❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥ ❞♦❡s ♥♦t ❝♦♥t❛✐♥ t❤❡ ✈❛❧✉❡ ②❂✸✳

❊①❝❡r❝✐s❡ ✶✳✻✳✶✸ ❋✐♥❞ t❤❡ ❞♦♠❛✐♥s ♦❢ t❤❡ ❢✉♥❝t✐♦♥s

f(x) =2x− 1

x2 − x, g(x) =

√5− x ❛♥❞ h(x) =

√

(x− 3)(5− x),

❙♦❧✉t✐♦♥✳ ❚❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❢ ✐s t❤❡ s❡t ♦❢ r❡❛❧s ❡①❝❡♣t t❤❡ r♦♦ts①❂✵ ♦r ✶ ♦❢ t❤❡ ❞❡♥♦♠✐♥❛t♦r✳ ■♥ ❝❛s❡ ♦❢ ❢✉♥❝t✐♦♥ ❣ ✇❡ ♥❡❡❞ t❤❡ s❡t ♦❢ r❡❛❧ss❛t✐s❢②✐♥❣

5− x ≥ 0,

✐✳❡✳ t❤❡ ❞♦♠❛✐♥ ✐s t❤❡ s❡t ♦❢ r❡❛❧s ❧❡ss ♦r ❡q✉❛❧ t❤❛♥ ✺✳ ❋✐♥❛❧❧② ✇❡ ❤❛✈❡ t♦s♦❧✈❡ t❤❡ ✐♥❡q✉❛❧✐t②

(x− 3)(5− x) ≥ 0.

❚❤❡ ❧❡❢t ❤❛♥❞ s✐❞❡ ✐s ♥♦♥✲♥❡❣❛t✐✈❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢

x− 3 ≥ 0 ❛♥❞ 5− x ≥ 0

♦rx− 3 ≤ 0 ❛♥❞ 5− x ≤ 0.

❚❤❡r❡❢♦r❡3 ≤ x ≤ 5.

❊①❝❡r❝✐s❡ ✶✳✻✳✶✹ ❋✐♥❞ t❤❡ ❞♦♠❛✐♥s ♦❢ t❤❡ ❢✉♥❝t✐♦♥s

f(x) =1

x+ 3, g(x) =

√2x+ 4 ❛♥❞ h(x) =

√

(x− 2)(x+ 3),

✹✵ ❈❍❆P❚❊❘ ✶✳ ●❊◆❊❘❆▲ ❈❖▼P❯❚❆❚■❖◆❆▲ ❙❑■▲▲❙

❙♦❧✉t✐♦♥✳ ❚❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❢ ✐s t❤❡ s❡t ♦❢ r❡❛❧s ❡①❝❡♣t ✲ ✸✳ ❋♦r t❤❡❢✉♥❝t✐♦♥ ❣ ✇❡ ❤❛✈❡

2x+ 4 ≥ 0 ⇒ x ≥ −2.

❋✐♥❛❧❧② ✇❡ ❤❛✈❡ t♦ s♦❧✈❡ t❤❡ ✐♥❡q✉❛❧✐t②

(x− 2)(x+ 3) ≥ 0.

❚❤❡ ❧❡❢t ❤❛♥❞ s✐❞❡ ✐s ♥♦♥✲♥❡❣❛t✐✈❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢

x− 2 ≥ 0 ❛♥❞ x+ 3 ≥ 0

♦rx− 2 ≤ 0 ❛♥❞ x+ 3 ≤ 0.

❚❤❡r❡❢♦r❡x ≤ −3 ♦r x ≥ 2.

❊①❝❡r❝✐s❡ ✶✳✻✳✶✺ ❊①♣r❡ss t❤❡ ♥✉♠❜❡rs

ln√3 ❛♥❞ ln

1

81

✐♥ t❡r♠s ♦❢ ln 3

❙♦❧✉t✐♦♥✳ ❙✐♥❝❡ √3 = 31/2 ❛♥❞

1

81= 3−4

✇❡ ❤❛✈❡ t❤❛t

ln√3 =

1

2ln 3

❛♥❞

ln1

81= −4 ln 3.

❊①❝❡r❝✐s❡ ✶✳✻✳✶✻ ❙♦❧✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥s

2x3x+2 = 54, 3x2x+2 = 24 ❛♥❞ ln(

x(x− 2)) = 0.

❙♦❧✉t✐♦♥✳ ❚♦ s♦❧✈❡ t❤❡ ✜rst ❡q✉❛t✐♦♥ ♦❜s❡r✈❡ t❤❛t

9 · 2x3x = 54,

6x = 6 ⇒ x = 1.

■♥ ❛ s✐♠✐❧❛r ✇❛②4 · 2x3x = 24,

6x = 6 ⇒ x = 1.

❋✐♥❛❧❧②x(x− 2) = 1,

x2 − 2x− 1 = 0 ⇒ x12 =2±

√4 + 4

2= 1±

√2.

✶✳✼✳ ▼❊❆◆❙ ✹✶

✶✳✼ ▼❡❛♥s

■♥ ♣r❛❝t✐❝❡ ❡st✐♠❛t✐♦♥s ❛r❡ ♦❢t❡♥ ♠♦r❡ ✐♠♣♦rt❛♥t t❤❛♥ t❤❡ ❡①❛❝t ✈❛❧✉❡s ♦❢q✉❛♥t✐t✐❡s✳ ▲♦ts ♦❢ ♥✉♠❡r✐❝❛❧ ✈❛❧✉❡s ❛r❡ ❢r❡q✉❡♥t❧② s✉❜st✐t✉t❡❞ ✇✐t❤ ♦♥❧② ♦♥❡❞✐st✐♥❣✉✐s❤❡❞ q✉❛♥t✐t② ❛s ✇❡ ❤❛✈❡ s❡❡♥ ❛❜♦✈❡ ✐♥ ❡①❡r❝✐s❡ ✶✳✻✳✽✳ ❚❤❡r❡ ❛r❡s❡✈❡r❛❧ r❡❛s♦♥s ✇❤② t♦ ✉s❡ ❛✈❡r❛❣❡ ✭♠❡❛♥✱ ♠♦❞❡✱ ♠❡❞✐❛♥✱ ❡❝♣❡❝t❛❜❧❡ ✈❛❧✉❡❡t❝✳✮ ✐♥ ♠❛t❤❡♠❛t✐❝s✳ ❆♥ ❛✈❡r❛❣❡ ✐s ❛ ♠❡❛s✉r❡ ♦❢ t❤❡ ♠✐❞❞❧❡ ♦r t②♣✐❝❛❧✈❛❧✉❡ ♦❢ ❛ ❞❛t❛ s❡t✳ ❚❤❡ ❣❡♥❡r❛❧ ❛✐♠ ✐s t♦ ❛❝❝✉♠✉❧❛t❡ t❤❡ ✐♥❢♦r♠❛t✐♦♥ ♦rt♦ s✉❜st✐t✉t❡ ♠♦r❡ ❝♦♠♣❧✐❝❛t❡❞ ♠❛t❤❡♠❛t✐❝❛❧ ♦❜❥❡❝ts ✇✐t❤ r❡❧❛t✐✈❡❧② s✐♠♣❧❡r♦♥❡s✳ ■♥ ✇❤❛t ❢♦❧❧♦✇s ✇❡ s✉♠♠❛r✐③❡ s♦♠❡ t❤❡♦r❡t✐❝❛❧ ♠❡t❤♦❞s t♦ ❝r❡❛t❡ ❛♥❛✈❡r❛❣❡✳

• ❚❤❡ ❛r✐t❤♠❡t✐❝ ♠❡❛♥ ♦❢ ❛ ✜♥✐t❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❞❛t❛ ✐s

A =x1 + x2 + . . .+ xn

n.

• ■♥ ❝❛s❡ ♦❢ ♥♦♥♥❡❣❛t✐✈❡ ♥✉♠❜❡rs ✇❡ ❝❛♥ ❢♦r♠ t❤❡ s♦✲❝❛❧❧❡❞ ❣❡♦♠❡tr✐❝♠❡❛♥

G = n√x1 · x2 · . . . · xn.

• ❚❤❡ ❤❛r♠♦♥✐❝ ♠❡❛♥ ♦❢ t❤❡ ❞❛t❛ s❡t ✐s

H =n

1

x1+ 1

x2+ . . .+ 1

xn

.

❘❡♠❛r❦ ❯s✐♥❣ ❚❤❛❧❡s t❤❡♦r❡♠ ✇❡ ❝❛♥ ✐♥t❡r♣r❡t t❤❡ ❛r✐t❤♠❡t✐❝ ♠❡❛♥ ♦❢①❂❆❋ ❛♥❞ ②❂❋❇ ❛s t❤❡ r❛❞✐✉s ♦❢ t❤❡ ❝✐r❝✉♠s❝r✐❜❡❞ ❝✐r❝❧❡ ♦❢ ❛ r✐❣❤t tr✐❛♥❣❧❡✇✐t❤ ❤②♣♦t❤❡♥✉s❡ ❆❇✳ ❚❤❡ ❤❡✐❣❤t ✐s ❥✉st t❤❡ ❣❡♦♠❡tr✐❝ ♠❡❛♥ ♦❢ ① ❛♥❞ ②✳❯♥❞❡r t❤❡ ❝❤♦✐❝❡ ①❂✶ ❛♥❞ ②❂♥ t❤✐s ❣✐✈❡s ❛♥ ❛❧t❡r♥❛t✐✈❡ ♠❡t❤♦❞ t♦ ❝♦♥str✉❝tt❤❡ r♦♦t ♦❢ ❛♥② ♥❛t✉r❛❧ ♥✉♠❜❡r ♥ ❜② r✉❧❡r ❛♥❞ ❝♦♠♣❛ss✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✜❣✉r❡ ✶✳✼ s❤♦✇s t❤❛t

G ≤ A

❢♦r t✇♦ ✈❛r✐❛❜❧❡s✳

■♥ ♠❛♥② s✐t✉❛t✐♦♥s ✐♥✈♦❧✈✐♥❣ r❛t❡s ❛♥❞ r❛t✐♦s t❤❡ ❤❛r♠♦♥✐❝ ♠❡❛♥ ♣r♦✲✈✐❞❡s t❤❡ tr✉❡st ❛✈❡r❛❣❡✳ ■❢ ❛ ✈❡❤✐❝❧❡ tr❛✈❡❧s ❛ ❝❡rt❛✐♥ ❞✐st❛♥❝❡ ❞ ❛t s♣❡❡❞ ✻✵❦✐❧♦♠❡tr❡s ♣❡r ❤♦✉r ❛♥❞ t❤❡♥ t❤❡ s❛♠❡ ❞✐st❛♥❝❡ ❛❣❛✐♥ ❛t s♣❡❡❞ ✹✵ ❦✐❧♦♠❡tr❡s♣❡r ❤♦✉r t❤❡♥ ✐ts ❛✈❡r❛❣❡ s♣❡❡❞ ✐s t❤❡ ❤❛r♠♦♥✐❝ ♠❡❛♥ ♦❢ ✻✵ ❛♥❞ ✹✵✱ ✐✳❡✳

21

60+ 1

40

= 48.

✹✷ ❈❍❆P❚❊❘ ✶✳ ●❊◆❊❘❆▲ ❈❖▼P❯❚❆❚■❖◆❆▲ ❙❑■▲▲❙

❋✐❣✉r❡ ✶✳✽✿ ❆r✐t❤♠❡t✐❝ ✈s✳ ❣❡♦♠❡tr✐❝ ♠❡❛♥s✳

■♥ ♦t❤❡r ✇♦r❞s t❤❡ t♦t❛❧ tr❛✈❡❧ t✐♠❡ ✐s t❤❡ s❛♠❡ ❛s ✐❢ t❤❡ ✈❡❤✐❝❧❡ ❤❛❞ tr❛✈❡❧❡❞t❤❡ ✇❤♦❧❡ ❞✐st❛♥❝❡ ❛t s♣❡❡❞ ✹✽ ❦✐❧♦♠❡tr❡s ♣❡r ❤♦✉r ❜❡❝❛✉s❡

d

t1= 60,

d

t2= 40

❛♥❞ t❤✉s2d

t1 + t2=

2dd60

+ d40

=2

1

60+ 1

40

.

❚❤❡ s❛♠❡ ♣r✐♥❝✐♣❧❡ ❝❛♥ ❜❡ ❛♣♣❧✐❡❞ t♦ ♠♦r❡ t❤❛♥ t✇♦ s❡❣♠❡♥ts ♦❢ t❤❡ ♠♦t✐♦♥✿✐❢ ✇❡ ❤❛✈❡ ❛ s❡r✐❡s ♦❢ s✉❜ ✲ tr✐♣s ❛t ❞✐✛❡r❡♥t s♣❡❡❞s ❛♥❞ ❡❛❝❤ s✉❜ ✲ tr✐♣❝♦✈❡rs t❤❡ s❛♠❡ ❞✐st❛♥❝❡ t❤❡♥ t❤❡ ❛✈❡r❛❣❡ s♣❡❡❞ ✐s t❤❡ ❤❛r♠♦♥✐❝ ♠❡❛♥ ♦❢ ❛❧❧t❤❡ s✉❜ ✲ tr✐♣ s♣❡❡❞s✳ ❆❢t❡r ❛ s❧✐❣❤t ♠♦❞✐✜❝❛t✐♦♥ ✇❡ ❝❛♥ ❣✐✈❡ t❤❡ ♣❤②s✐❝❛❧✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ t❤❡ ❛r✐t❤♠❡t✐❝ ♠❡❛♥ t♦♦✿ ✐❢ ❛ ✈❡❤✐❝❧❡ tr❛✈❡❧s ❢♦r ❛ ❝❡rt❛✐♥❛♠♦✉♥t ✧t✧ ♦❢ t✐♠❡ ❛t s♣❡❡❞ ✻✵ ❛♥❞ t❤❡♥ t❤❡ s❛♠❡ ❛♠♦✉♥t ♦❢ t✐♠❡ ❛t s♣❡❡❞✹✵ t❤❡♥ t❤❡ ❛✈❡r❛❣❡ s♣❡❡❞ ✐s ❥✉st t❤❡ ❛r✐t❤♠❡t✐❝ ♠❡❛♥ ♦❢ ✻✵ ❛♥❞ ✹✵✱ ✐✳❡✳

60 + 40

2= 50.

■♥ ♦t❤❡r ✇♦r❞s t❤❡ t♦t❛❧ ❞✐st❛♥❝❡ ✐s t❤❡ s❛♠❡ ❛s ✐❢ t❤❡ ✈❡❤✐❝❧❡ ❤❛❞ tr❛✈❡❧❡❞❢♦r t❤❡ ✇❤♦❧❡ t✐♠❡ ❛t s♣❡❡❞ ✺✵ ❦✐❧♦♠❡tr❡s ♣❡r ❤♦✉r ❜❡❝❛✉s❡

s1t= 60,

s2t= 40

❛♥❞ t❤✉ss1 + s2

2t=

60 + 40

2.

✶✳✽✳ ❊❳❊❘❈■❙❊❙ ✹✸

✶✳✽ ❊①❡r❝✐s❡s

❊①❝❡r❝✐s❡ ✶✳✽✳✶ ❋✐♥❞ t❤❡ ❛r✐t❤♠❡t✐❝ ♠❡❛♥ ♦❢ t❤❡ ♣♦ss✐❜❧❡ ✈❛❧✉❡s ♦❢ t❤❡ ♣❛✲r❛♠❡t❡r ✧❛✧ ✐♥ ❡①❡r❝✐s❡ ✶✳✻✳✽✳

❙♦❧✉t✐♦♥✳

A = −0.18 + 0.21 + 0.25 + 0.32 + 0.4 + 0.2 + 0.17 + 0.21 + 0.18 + 0.18

11=

−0.2.

❊①❝❡r❝✐s❡ ✶✳✽✳✷ Pr♦✈❡ t❤❛t ❢♦r ❛♥② ♣❛✐r ♦❢ ♣♦s✐t✐✈❡ r❡❛❧ ♥✉♠❜❡rs ① ❛♥❞ ②

21

x+ 1

y

≤ √xy ≤ x+ y

2

✳

❙♦❧✉t✐♦♥✳ ❆t ✜rst ✇❡ ♣r♦✈❡ t❤❛t ❢♦r ❛♥② ♣❛✐r ♦❢ ♥♦♥✲♥❡❣❛t✐✈❡ ♥✉♠❜❡rs ① ❛♥❞②

√xy ≤ x+ y

2.

❚❛❦✐♥❣ t❤❡ sq✉❛r❡ ♦❢ ❜♦t❤ s✐❞❡s ✇❡ ❤❛✈❡ t❤❛t

xy ≤ (x+ y)2

4

❛♥❞✱ ❝♦♥s❡q✉❡♥t❧②✱4xy ≤ x2 + 2xy + y2,

0 ≤ x2 − 2xy + y2 = (x− y)2

✇❤✐❝❤ ✐s ♦❜✈✐♦✉s❧② tr✉❡✳ ■❢ ❛❂✶✴① ❛♥❞ ❜❂✶✴② t❤❡♥

√ab ≤ a+ b

2

✇❤✐❝❤ ♠❡❛♥s t❤❛t2

1

x+ 1

y

=2

a+ b≤ 1√

ab=

√xy

❛s ✇❛s t♦ ❜❡ ♣r♦✈❡❞✳

❊①❝❡r❝✐s❡ ✶✳✽✳✸ ❙✉♣♣♦s❡ t❤❛t ②♦✉ ✇❛♥t t♦ ❝r❡❛t❡ ❛ r❡❝t❛♥❣✉❧❛r✲s❤❛♣❡❞ ❣❛r✲❞❡♥ ✇✐t❤ ❛r❡❛ ✶✵✷✹ sq✉❛r❡ ❢♦♦t❛❣❡✳ ❍♦✇ ♠❛♥② ❢❡❡t ✐♥ ❧❡♥❣t❤ ②♦✉ ♥❡❡❞ t♦ ❢❡♥❝❡②♦✉r ❣❛r❞❡♥❄

✹✹ ❈❍❆P❚❊❘ ✶✳ ●❊◆❊❘❆▲ ❈❖▼P❯❚❆❚■❖◆❆▲ ❙❑■▲▲❙

❍✐♥t✳ ❚❤❡ ♣r♦❜❧❡♠ ✐s t♦ ♠✐♥✐♠✐③❡ t❤❡ ♣❡r✐♠❡t❡r ❛♠♦♥❣ r❡❝t❛♥❣❧❡s ✇✐t❤ ❛r❡❛✶✵✷✹✳ ▲❡t ① ❛♥❞ ② ❜❡ t❤❡ s✐❞❡s ♦❢ ❛ r❡❝t❛♥❣❧❡✳ ❚♦ ✜♥❞ t❤❡ ♠✐♥✐♠✉♠ ♦❢ t❤❡♣❡r✐♠❡t❡r ✷✭①✰②✮ s✉❜❥❡❝t t♦ t❤❡ ❡q✉❛❧✐t② ❝♦♥str❛✐♥ ①②❂✶✵✷✹ ✐♥tr♦❞✉❝❡ t❤❡❢✉♥❝t✐♦♥

f(x, y) = 2(x+ y).

❙✉❜st✐t✉t✐♥❣ ②❂✶✵✷✹✴① ✇❡ ❝❛♥ r❡❞✉❝❡ t❤❡ ♥✉♠❜❡r ♦❢ ✈❛r✐❛❜❧❡s✿

f(x) = 2

(

x+1024

x

)

.

❚❤❡ r❡❧❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ t❤❡ ❛r✐t❤♠❡t✐❝ ❛♥❞ t❤❡ ❣❡♦♠❡tr✐❝ ♠❡❛♥s s❤♦✇s t❤❛t

f(x) = 2

(

x+1024

x

)

= 4x+ (1024/x)

2≥ 4

√1024 = 128

❛♥❞ ❡q✉❛❧✐t② ❤❛♣♣❡♥s ✐❢ ❛♥❞ ♦♥❧② ✐❢ ①❂✸✷✳ ❚❤✐s ✐s t❤❡ ❝❛s❡ ♦❢ t❤❡ sq✉❛r❡✳

❊①❝❡r❝✐s❡ ✶✳✽✳✹ ❋♦r♠✉❧❛t❡ t❤❡ ♣❤②s✐❝❛❧ ♣r✐♥❝✐♣❧❡ ❢♦r t❤❡ ❛r✐t❤♠❡t✐❝ ♠❡❛♥✳

❙♦❧✉t✐♦♥✳ ■❢ ✇❡ ❤❛✈❡ ❛ s❡r✐❡s ♦❢ s✉❜ ✲ tr✐♣s ❛t ❞✐✛❡r❡♥t s♣❡❡❞s ❛♥❞ ❡❛❝❤ s✉❜ ✲tr✐♣ t❛❦❡s t❤❡ s❛♠❡ ❛♠♦✉♥t ♦❢ t✐♠❡ t❤❡♥ t❤❡ ❛✈❡r❛❣❡ s♣❡❡❞ ✐s t❤❡ ❛r✐t❤♠❡t✐❝♠❡❛♥ ♦❢ ❛❧❧ t❤❡ s✉❜ ✲ tr✐♣ s♣❡❡❞s✳

✶✳✾ ❊q✉❛t✐♦♥s✱ s②st❡♠ ♦❢ ❡q✉❛t✐♦♥s

❚❤❡ ♠❛t❤❡♠❛t✐❝❛❧ ❢♦r♠✉❧❛t✐♦♥ ♦❢ ♣r♦❜❧❡♠s ♦❢t❡♥ ❣✐✈❡s ❛ s✐♥❣❧❡ ❡q✉❛t✐♦♥ ♦rs②st❡♠ ♦❢ ❡q✉❛t✐♦♥s ✭s❡❡ ❡✳❣✳ ❝♦♦r❞✐♥❛t❡ ❣❡♦♠❡tr②✮✳ ■t ✐s ✐♠♣♦rt❛♥t t♦ ✐s♦❧❛t❡r❡❧❡✈❛♥t ✐♥❢♦r♠❛t✐♦♥✿

❆ r❡❝t❛♥❣✉❧❛r ❜♦① ✇✐t❤ ❛ ❜❛s❡ ✷ ✐♥❝❤❡s ❜② ✻ ✐♥❝❤❡s ✐s ✶✵ ✐♥❝❤❡s t❛❧❧❛♥❞ ❤♦❧❞s ✶✷ ♦✉♥❝❡s ♦❢ ❜r❡❛❦❢❛st ❝❡r❡❛❧✳ ❚❤❡ ♠❛♥✉❢❛❝t✉r❡r ✇❛♥tst♦ ✉s❡ ❛ ♥❡✇ ❜♦① ✇✐t❤ ❛ ❜❛s❡ ✸ ✐♥❝❤❡s ❜② ✺ ✐♥❝❤❡s✳ ❍♦✇ ♠❛♥②✐♥❝❤❡s t❛❧❧ s❤♦✉❧❞ ❜❡ ✐♥ ♦r❞❡r t♦ ❤♦❧❞ ❡①❛❝t❧② t❤❡ s❛♠❡ ✈♦❧✉♠❡ ❛st❤❡ ♦r✐❣✐♥❛❧ ❜♦①❄

r❡❧❡✈❛♥t ✐♥❢♦r♠❛t✐♦♥ ✐rr❡❧❡✈❛♥t ✐♥❢♦r♠❛t✐♦♥t❤❡ ❜❛s❡ ✐s 2× 6 ✐♥❝❤t❤❡ t❛❧❧ ✐s 10 ✐♥❝❤✱ 12 ♦✉♥❝❡s ♦❢ ❜r❡❛❦❢❛st ❝❡r❡❛❧

t❤❡ ♥❡✇ ❜❛s❡ ✐s 3× 5 ♠❛♥✉❢❛❝t✉r❡r✱ ✐♥❝❤❞♦♥✬t ❝❤❛♥❣❡ t❤❡ ✈♦❧✉♠❡ ✲

✶✳✶✵✳ ❊❳❊❘❈■❙❊❙ ✹✺

❋✐❣✉r❡ ✶✳✾✿ ❊①❡r❝✐s❡ ✶✳✶✵✳✶

❚❤❡ ♦♥❧② t❤❡♦r❡t✐❝❛❧ ❢❛❝t ✇❡ ♥❡❡❞ t♦ s♦❧✈❡ t❤❡ ♣r♦❜❧❡♠ ✐s t❤❛t t❤❡ ✈♦❧✉♠❡♦❢ ❛ r❡❝t❛♥❣✉❧❛r ❜♦① ✐s ❥✉st t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ ❛r❡❛ ♦❢ t❤❡ ❜❛s❡ ❛♥❞ t❤❡ t❛❧❧✳❚❤❡r❡❢♦r❡ ✇❡ ❝❛♥ ✇r✐t❡ t❤❡ ❡q✉❛t✐♦♥

2 · 6 · 10 = 3 · 5 ·m,

✇❤❡r❡ ♠ ❞❡♥♦t❡s t❤❡ ✉♥❦♥♦✇♥ t❛❧❧ ✭❤❡✐❣❤t✮ ♦❢ t❤❡ ♥❡✇ ❜♦①✳ ❲❡ ❤❛✈❡ t❤❛t♠❂✽✳ ◗✉❛♥t✐t✐❡s ✇❡ ❛r❡ ❧♦♦❦✐♥❣ ❢♦r ♠❛② ❤❛✈❡ ❛ ♠♦r❡ ❝♦♠♣❧✐❝❛t❡❞ r❡❧❛t✐♦♥s❤✐♣✇✐t❤ t❤❡ ❣✐✈❡♥ ❞❛t❛✳ ❙♦♠❡t✐♠❡s ✇❡ s❤♦✉❧❞ ✇r✐t❡ ♠♦r❡ t❤❛♥ ♦♥❡ r❡❧❛t✐♦♥s❤✐♣s✭t♦❣❡t❤❡r ✇✐t❤ ♥❡✇ ❛✉①✐❧✐❛r② ✈❛r✐❛❜❧❡s✮ t♦ ❝♦♠♣✉t❡ t❤❡ ♠✐ss✐♥❣ ♦♥❡✳

✶✳✶✵ ❊①❡r❝✐s❡s

❊①❝❡r❝✐s❡ ✶✳✶✵✳✶ ■♥ r❡❝t❛♥❣❧❡ ❆❇❈❉✱ s✐❞❡ ❆❇ ✐s t❤r❡❡ t✐♠❡s ❧♦♥❣❡r t❤❛♥❇❈✳ ❚❤❡ ❞✐st❛♥❝❡ ♦❢ ❛♥ ✐♥t❡r✐♦r ♣♦✐♥t P ❢r♦♠ t❤❡ ✈❡rt✐❝❡s ❆✱ ❇ ❛♥❞ ❉ ❛r❡

PA =√2, PB = 4

√2 ❛♥❞ PD = 2,

r❡s♣❡❝t✐✈❡❧②✳ ❲❤❛t ✐s t❤❡ ❛r❡❛ ♦❢ t❤❡ r❡❝t❛♥❣❧❡✳

❍✐♥t✳ ❯s✐♥❣ ♦rt❤♦❣♦♥❛❧ ♣r♦❥❡❝t✐♦♥s ♦❢ t❤❡ ✐♥t❡r✐♦r ♣♦✐♥t P t♦ t❤❡ s✐❞❡s ♦❢ t❤❡r❡❝t❛♥❣❧❡ ✇❡ ❝❛♥ ✉s❡ P②t❤❛❣♦r❡❛♥ t❤❡♦r❡♠ t❤r❡❡ t✐♠❡s✿

AQ2 + AR2 = 2,

(AD − AQ)2 + AR2 = 4,

(AB − AR)2 + AQ2 = (4√2)2 = 32.

✹✻ ❈❍❆P❚❊❘ ✶✳ ●❊◆❊❘❆▲ ❈❖▼P❯❚❆❚■❖◆❆▲ ❙❑■▲▲❙

❙✐♥❝❡ ❆❇❂✸❆❉ ✇❡ ❤❛✈❡ t❤r❡❡ ❡q✉❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐t✐❡s ①❂❆❉✱ ②❂❆◗❛♥❞ ③❂❆❘✳ ◆❛♠❡❧②

y2 + z2 = 2,

(x− y)2 + z2 = 4,

(3x− z)2 + y2 = 32.

❲❡ ❤❛✈❡ t❤❛t

4 = (x− y)2 + z2 = x2 − 2xy + y2 + z2 = x2 − 2xy + 2

❛♥❞

32 = (3x− z)2 + y2 = 9x2 − 6xz + z2 + y2 = 9x2 − 6xz + 2.

❚❤❡r❡❢♦r❡

y =x2 − 2

2x, z =

9x2 − 30

6x❛♥❞ t❤❡ ✜rst ❡q✉❛t✐♦♥ ❣✐✈❡s t❤❛t

(a− 2)2

4a+

(9a− 30)2

36a= 2,

✇❤❡r❡ ❛❂x2✳ ❋r♦♠ ❤❡r❡

9(a− 2)2 + (9a− 30)2 = 72a,

90a2 − 648a+ 936 = 0.

❋✐♥❛❧❧②5a2 − 36a+ 52 = 0

✇❤✐❝❤ ♠❡❛♥s t❤❛t

a12 =36±

√256

10⇒ a = 2 ♦r 5.2.

■❢ ❛❂✷ t❤❡♥ ✇❡ ❤❛✈❡ t❤❛t

x2 = 2 ⇒ y = 0 ❛♥❞ z < 0

✇❤✐❝❤ ✐s ✐♠♣♦ss✐❜❧❡✳ ❚❤❡r❡❢♦r❡

x2 = 5.2 ⇒ A = 3x2 = 15.6.

◆♦t❡ t❤❛t t❤❡r❡ ✐s ♥♦ ♥❡❡❞ t♦ ❝♦♠♣✉t❡ ① ❜❡❝❛✉s❡ t❤❡ ❛r❡❛ ♦❢ t❤❡ r❡❝t❛♥❣❧❡❝❛♥ ❜❡ ❣✐✈❡♥ ❛s 3x2 = 3a✳

❘❡♠❛r❦ ❙②st❡♠s ❝♦♥t❛✐♥✐♥❣ q✉❛❞r❛t✐❝ ❡q✉❛t✐♦♥s ❛r❡ t②♣✐❝❛❧ ✐♥ ❝♦♦r❞✐♥❛t❡❣❡♦♠❡tr②✿ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ ❛ ❧✐♥❡ ❛♥❞ ❛ ❝✐r❝❧❡ ♦r t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t✇♦❝✐r❝❧❡s✳

❈❤❛♣t❡r ✷

❊①❡r❝✐s❡s

✷✳✶ ❊①❡r❝✐s❡s

❊①❝❡r❝✐s❡ ✷✳✶✳✶ ❲✐t❤♦✉t ❝❛❧❝✉❧❛t♦r ✜♥❞ t❤❡ ✈❛❧✉❡s ♦❢

82

3 · 2−2, 77782 − 22232,4372 − 3632

5372 − 4632,

√

5− 2√6 +

√

3− 2√2 +

√

7− 2√12,

(

1 +1

2

)

·(

1 +1

3

)

·(

1 +1

4

)

· . . . ·(

1 +1

100

)

.

❙♦❧✉t✐♦♥✳ ❯s✐♥❣ ♣♦✇❡r ❧❛✇ ✐❞❡♥t✐t✐❡s

82

3 · 2−2 =(

81

3

)2

· 1

22= 22 · 1

22= 1.

❙❡❝♦♥❞❧②

77782 − 22232 = (7778− 2223)(7778 + 2223) = 5555 · 10001 =

= 5555(10000 + 1) = 55550000 + 5555 = 55555555.

■♥ t❤❡ s❛♠❡ ✇❛②

4372 − 3632 = (437− 363)(437 + 363) = 74 · 800,

5372 − 4632 = (537− 463)(537 + 463) = 74 · 1000❛♥❞✱ ❝♦♥s❡q✉❡♥t❧②✱

4372 − 3632

5372 − 4632= 0.8.

✹✼

✹✽ ❈❍❆P❚❊❘ ✷✳ ❊❳❊❘❈■❙❊❙

❚♦ ❝♦♠♣✉t❡ t❤❡ ❡①❛❝t ✈❛❧✉❡s ♦❢ t❤❡ r♦♦ts ♥♦t❡ t❤❛t

(√3−

√2)2 = 3− 2

√6 + 2 = 5− 2

√6.

❚❤❡r❡❢♦r❡√

5− 2√6 =

√3−

√2.

■♥ ❛ s✐♠✐❧❛r ✇❛②√

3− 2√2 =

√2− 1,

√

7− 2√12 = 2−

√3.

❚❤❡r❡❢♦r❡√

5− 2√6 +

√

3− 2√2 +

√

7− 2√12 = 1.

❖❜s❡r✈❡ t❤❛t

1 +1

n=

n+ 1

n

❛♥❞ t❤✉s(

1 +1

2

)

·(

1 +1

3

)

·(

1 +1

4

)

· . . . ·(

1 +1

100

)

=

3

2· 43· 54· . . . · 100

90· 101100

=101

2.

❊①❝❡r❝✐s❡ ✷✳✶✳✷ ❙♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥s✿

3x− 4

3=

5

12, x2 − x− 6 = 0, x3 + 6x2 − 4x− 24 = 0,

1

x2 − 9+

1

x− 3=

48

(x− 3)(x+ 38).

❙♦❧✉t✐♦♥✳ ❚❤❡ ✜rst ❡q✉❛t✐♦♥ s❛②s t❤❛t

3x =5

12+

4

3,

✐✳❡✳ ✸①❂✷✶✴✶✷ ❛♥❞ t❤✉s ①❂✼✴✶✷✳ ❙❡❝♦♥❞❧②

x12 =1±

√1 + 4 · 62

=1± 5

2.

❯s✐♥❣ t❤❡ t❡❝❤♥✐❝ ♦❢ ❞✐✈✐s✐♦♥ ♦❢ ♣♦❧②♥♦♠✐❛❧s ✐t ❝❛♥ ❜❡ ❡❛s✐❧② s❡❡♥ t❤❛t ✐❢ ❛♣♦❧②♥♦♠✐❛❧ ❤❛s ❛♥ ✐♥t❡❣❡r r♦♦t ♠ t❤❡♥ ✐t ♠✉st ❞✐✈✐❞❡ t❤❡ ❝♦♥st❛♥t t❡r♠✳ ❲❡

✷✳✶✳ ❊❳❊❘❈■❙❊❙ ✹✾

❛r❡ ❣♦✐♥❣ t♦ ❣✉❡ss ❛t ❧❡❛st ♦♥❡ ♦❢ t❤❡ r♦♦ts ♦❢ t❤❡ ♣♦❧②♥♦♠✐❛❧ ❜② ❝❤❡❝❦✐♥❣t❤❡ ❞✐✈✐s♦rs ♦❢ ✷✹✳ ❚❤✐s r❡s✉❧ts ✐♥ t❤❡ r♦♦t ♠❂✷✳ ❯s✐♥❣ ♣♦❧②♥♦♠✐❛❧ ❞✐✈✐s✐♦♥❛❣❛✐♥

x3 + 6x2 − 4x− 24 = (x− 2)(x2 + 8x+ 12).

❚♦ ✜♥✐s❤ t❤❡ s♦❧✉t✐♦♥ ✇❡ s♦❧✈❡ t❤❡ q✉❛❞r❛t✐❝ ❡q✉❛t✐♦♥

x2 + 8x+ 12 = 0

t♦♦✳ ❲❡ ❤❛✈❡

x12 =−8±

√82 − 4 · 122

=−8± 4

2= −4± 2.

■♥ ❝❛s❡ ♦❢ t❤❡ ❧❛st ❡q✉❛t✐♦♥ ✇❡ ✉s❡ t❤❡ ✐❞❡♥t✐t② x2 − 9 = (x − 3)(x + 3) t♦❝♦♥❝❧✉❞❡ t❤❛t

1

x+ 3+ 1 =

48

x+ 38

✇❤✐❝❤ r❡s✉❧ts ✐♥ ❛ q✉❛❞r❛t✐❝ ❡q✉❛t✐♦♥✳

❊①❝❡r❝✐s❡ ✷✳✶✳✸ Pr♦✈❡ t❤❛t

√

6 +

√

6 +

√

6 +√6 < 3.

❙♦❧✉t✐♦♥✳ ❚❛❦✐♥❣ t❤❡ sq✉❛r❡ ♦❢ ❜♦t❤ s✐❞❡s s②st❡♠❛t✐❝❛❧❧②

√

6 +

√

6 +√6 < 3,

√

6 +√6 < 3,

√6 < 3,

6 < 9

✇❤✐❝❤ ✐s ♦❜✈✐♦✉s❧② tr✉❡✳

❊①❝❡r❝✐s❡ ✷✳✶✳✹ ❲❤✐❝❤ ♥✉♠❜❡r ✐s t❤❡ ❜✐❣❣❡r❄

297 · 299 ♦r 2982, 3452 ♦r 342 · 348,√101−

√100 ♦r

1

20.

✺✵ ❈❍❆P❚❊❘ ✷✳ ❊❳❊❘❈■❙❊❙

❙♦❧✉t✐♦♥✳ ❯s✐♥❣ t❤❛t

297 · 299 = (298− 1)(298 + 1) = 2982 − 1

✐t ❢♦❧❧♦✇s t❤❛t t❤❡ s❡❝♦♥❞ ♥✉♠❜❡r ✐s t❤❡ ❜✐❣❣❡r ♦♥❡✳ ■♥ ❛ s✐♠✐❧❛r ✇❛②

342 · 348 = (345− 3)(345 + 3) = 3452 − 32

❛♥❞ 3452 ✐s ❜✐❣❣❡r t❤❛♥ t❤❡ ♣r♦❞✉❝t 342 · 348✳ ❙✐♥❝❡

√101−

√100 =

(√101−

√100

)

√101 +

√100√

101 +√100

=1√

101 +√100

✐t ✐s ❡♥♦✉❣❤ t♦ ❝♦♠♣❛r❡ t❤❡ ♥✉♠❜❡rs√101 +

√100 ❛♥❞ 20.

❍❡r❡ √101 +

√100 >

√100 +

√100 = 20

✇❤✐❝❤ ♠❡❛♥s t❤❛t

√101−

√100 =

1√101 +

√100

<1

20.

❊①❝❡r❝✐s❡ ✷✳✶✳✺ ❙♦❧✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠s ♦❢ ❡q✉❛t✐♦♥s

3x− 7y = 66

2x− 9y = −8

❛♥❞x2 − y = 46

x2y = 147.

❙♦❧✉t✐♦♥✳ ■♥ t❡r♠s ♦❢ ❝♦♦r❞✐♥❛t❡ ❣❡♦♠❡tr② t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ✜rst s②st❡♠ ♦❢❡q✉❛t✐♦♥s ❣✐✈❡s t❤❡ ❝♦♠♠♦♥ ♣♦✐♥t ♦❢ t✇♦ ❧✐♥❡s✳ ❋r♦♠ t❤❡ ✜rst ❡q✉❛t✐♦♥ ✇❡❝❛♥ ✇r✐t❡ ② ✐♥ t❡r♠s ♦❢ ① ❛s ❢♦❧❧♦✇s

y =3x− 66

7.

❙✉❜st✐t✉t✐♥❣ t❤✐s ❡①♣r❡ss✐♦♥ ✐♥t♦ t❤❡ s❡❝♦♥❞ ❡q✉❛t✐♦♥ ✇❡ ❤❛✈❡ t❤❛t

2x− 93x− 66

7= −8,

✷✳✶✳ ❊❳❊❘❈■❙❊❙ ✺✶

14x− 9(3x− 66) = −56,

14x− 27x+ 594 = −56.

❋✐♥❛❧❧②

x =650

13= 50

❛♥❞✱ ❝♦♥s❡q✉❡♥t❧②✱

y =150− 66

7= 12.

❚♦ s♦❧✈❡ t❤❡ s❡❝♦♥❞ s②st❡♠ ♦❢ ❡q✉❛t✐♦♥s ✐t s❡❡♠s t♦ ❜❡ ♠♦r❡ ❝♦♥✈❡♥✐❡♥t t♦❡①♣r❡ss x2 ❢r♦♠ t❤❡ ✜rst ❡q✉❛t✐♦♥ ❛s ❢♦❧❧♦✇s

x2 = 46 + y.

❇② s✉❜st✐t✉t✐♦♥(46 + y)y = 147,

y2 + 46y − 147 = 0

✇❤✐❝❤ ♠❡❛♥s t❤❛t

y12 =−46±

√462 + 4 · 1472

=−46±

√2704

2=

−46± 52

2= −23± 26.

■❢ y = 3 t❤❡♥ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✈❛❧✉❡s ♦❢ ① ❛r❡ x = ±7✳ ■❢ y = −49 t❤❡♥t❤❡r❡ ✐s ♥♦ ❛♥② ❝♦rr❡s♣♦♥❞✐♥❣ ✈❛❧✉❡ ♦❢ ①✳

❊①❝❡r❝✐s❡ ✷✳✶✳✻ ❯s✐♥❣ ✐♥❞✉❝t✐♦♥ ♣r♦✈❡ t❤❛t

13 + 23 + 33 + . . .+ n3 =

(

n(n+ 1)

2

)2

✭✷✳✶✮

❙♦❧✉t✐♦♥✳ ■♥ ❝❛s❡ ♦❢ ♥❂✶ t❤❡ st❛t❡♠❡♥t ✐s ♦❜✈✐♦✉s❧② tr✉❡✳ ❯s✐♥❣ t❤❡ ✐♥❞✉❝t✐✈❡❤②♣♦t❤❡s✐s

13 + 23 + 33 + . . .+ n3 + (n+ 1)3 =

(

n(n+ 1)

2

)2

+ (n+ 1)3 =

=n2(n+ 1)2 + 4(n+ 1)3

22=

(n+ 1)2(n2 + 4(n+ 1))

22=

(n+ 1)2(n+ 2)2

22=

(n+ 1)2((n+ 1) + 1)2

22=

(

(n+ 1)((n+ 1) + 1)

2

)2

❛s ✇❛s t♦ ❜❡ ♣r♦✈❡❞✳

✺✷ ❈❍❆P❚❊❘ ✷✳ ❊❳❊❘❈■❙❊❙

❊①❝❡r❝✐s❡ ✷✳✶✳✼ ❈♦♠♣✉t❡ t❤❡ ✈❛❧✉❡s ♦❢

f(−4), f(−3) ❛♥❞ f(2)

❛♥❞ s❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥

f(x) = −1

2x2 − x+

3

2.

❙♦❧✉t✐♦♥✳ ❲❡ ❤❛✈❡

f(−4) = −1

2(−4)2 − (−4) +

3

2=

11

2,

f(−3) = −1

2(−3)2 − (−3) +

3

2= 0,

f(2) = −1

2(2)2 − 2 +

3

2= −5

2.

❚♦ s❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❝♦♥s✐❞❡r t❤❡ ❝❛♥♦♥✐❝❛❧ ❢♦r♠

f(x) = −1

2(x+ 1)2 + 2.

❚❤❡r❡❢♦r❡ t❤❡ ③❡r♦s ♦❢ t❤❡ ❢✉♥❝t✐♦♥ s❛t✐s❢② t❤❡ ❡q✉❛t✐♦♥

(x+ 1)2 = 4

✇❤✐❝❤ ♠❡❛♥s t❤❛t x1 = 1 ❛♥❞ x2 = −3✳ ❚❤❡ ♠❛①✐♠✉♠ ✈❛❧✉❡ ✐s ❥✉st ✷ ❛tt❛✐♥❡❞❛t t❤❡ ❛r✐t❤♠❡t✐❝ ♠❡❛♥ ♦❢ t❤❡ ③❡r♦s✿

x ♠❛① =1 + (−3)

2= −1.

❊①❝❡r❝✐s❡ ✷✳✶✳✽ ❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f(x) = x2 − 8x+ 15✳

❙♦❧✉t✐♦♥✳ ❚♦ s❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❝♦♥s✐❞❡r t❤❡ ❝❛♥♦♥✐❝❛❧ ❢♦r♠

f(x) = (x− 4)2 − 1.

❚❤❡ ③❡r♦s ♦❢ t❤❡ ❢✉♥❝t✐♦♥ s❛t✐s❢② t❤❡ ❡q✉❛t✐♦♥

(x− 4)2 = 1

✇❤✐❝❤ ♠❡❛♥s t❤❛t x1 = 3 ❛♥❞ x2 = 5✳ ❚❤❡ ♠✐♥✐♠✉♠ ✈❛❧✉❡ ✐s ❥✉st ✲ ✶ ❛tt❛✐♥❡❞❛t t❤❡ ❛r✐t❤♠❡t✐❝ ♠❡❛♥ ♦❢ t❤❡ ③❡r♦s✿

x ♠✐♥ =3 + 5

2= 4.

✷✳✶✳ ❊❳❊❘❈■❙❊❙ ✺✸

❋✐❣✉r❡ ✷✳✶✿ ❊①❡r❝✐s❡ ✷✳✶✳✼

❋✐❣✉r❡ ✷✳✷✿ ❊①❡r❝✐s❡ ✷✳✶✳✽

✺✹ ❈❍❆P❚❊❘ ✷✳ ❊❳❊❘❈■❙❊❙

❊①❝❡r❝✐s❡ ✷✳✶✳✾ ❋✐♥❞ ❛❧❧ ✐♥t❡❣❡r r♦♦ts ♦❢ t❤❡ ❡q✉❛t✐♦♥ 2x3+11x2−7x−6 = 0❛♥❞ ♣❡r❢♦r♠ t❤❡ ❞✐✈✐s✐♦♥

(2x3 + 2x− 1) : (x− 1) =

❙♦❧✉t✐♦♥✳ ❆♥② ✐♥t❡❣❡r r♦♦t ♠✉st ❜❡ ❛ ❞✐✈✐s♦r ♦❢ t❤❡ ❝♦♥st❛♥t t❡r♠✳ ❚❤❡r❡❢♦r❡t❤❡ ♣♦ss✐❜❧❡ ✈❛❧✉❡s ❛r❡

±1, ±2, ±3,±6.

❙✉❜st✐t✉t✐♥❣ t❤❡s❡ ✈❛❧✉❡s ❛s ① ✇❡ ❤❛✈❡ t❤❛t t❤❡ ✐♥t❡❣❡r r♦♦ts ❛r❡ ①❂✶ ♦r ✲ ✻✳❋✐♥❛❧❧②

2x3 + 11x2 − 7x− 6 ✿ x− 1 = 2x2

− (2x3 − 2x2)

13x2 − 7x− 6 ✿ x− 1 ❂13x− (13x2 − 13x)

6x− 6 ✿ x− 1 = 6− (6x− 6)

0

❚❤❡r❡❢♦r❡2x3 + x2 − 1 = (2x2 + 13x+ 6)(x− 1).

❚❤❡ ♠✐ss✐♥❣ r♦♦ts ❛r❡

x12 =−13±

√132 − 4 · 2 · 64

=−13± 11

4,

✐✳❡✳ x1 = −6 ❛♥❞ x2 = −(1/2)✳

❊①❝❡r❝✐s❡ ✷✳✶✳✶✵ ❙♦❧✈❡ t❤❡ ✐♥❡q✉❛❧✐t②

x2 − x− 6 < 0.

❙♦❧✉t✐♦♥✳ ❚❤❡ st❛♥❞❛r❞ ✇❛② ♦❢ s♦❧✈✐♥❣ q✉❛❞r❛t✐❝ ✐♥❡q✉❛❧✐t✐❡s ❝♦♥s✐sts ♦❢ t❤r❡❡st❡♣s✳ ❆t ✜rst ✇❡ ❞❡t❡r♠✐♥❡ t❤❡ r♦♦ts ♦❢ t❤❡ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧ ✐❢ ❡①✐st✿

x12 =1±

√1 + 24

2=

1± 5

2,

✐✳❡✳ x1 = −2 ❛♥❞ x2 = 3✳ ❙❡❝♦♥❞❧② ✇❡ s❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳ ❙✐♥❝❡t❤❡ ❝♦❡✣❝✐❡♥t ♦❢ t❤❡ t❡r♠ ♦❢ ❤✐❣❤❡st ❞❡❣r❡❡ ✐s ♣♦s✐t✐✈❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣♣❛r❛❜♦❧❛ ✐s ♦♣❡♥ ❢r♦♠ ❛❜♦✈❡ ✭✐♥ ♦t❤❡r ✇♦r❞s ✐t ❤❛s ❛ ♠✐♥✐♠✉♠ ❛tt❛✐♥❡❞ ❛tt❤❡ ❛r✐t❤♠❡t✐❝ ♠❡❛♥ ♦❢ t❤❡ r♦♦ts✮✳ ❋✐♥❛❧❧② t❤❡ s♦❧✉t✐♦♥s ❛r❡ ✲ ✷ ❁ ① ❁ ✸✳

✷✳✶✳ ❊❳❊❘❈■❙❊❙ ✺✺

❋✐❣✉r❡ ✷✳✸✿ ❊①❡r❝✐s❡ ✷✳✶✳✶✵✳

❊①❝❡r❝✐s❡ ✷✳✶✳✶✶ ❙♦❧✈❡ t❤❡ ✐♥❡q✉❛❧✐t②

x2 − x− 6 > 0.

✺✻ ❈❍❆P❚❊❘ ✷✳ ❊❳❊❘❈■❙❊❙

❈❤❛♣t❡r ✸

❇❛s✐❝ ❢❛❝ts ✐♥ ❣❡♦♠❡tr②

❯s✐♥❣ P❧❛t♦✬s ✇♦r❞s ✧t❤❡ ♦❜❥❡❝ts ♦❢ ❣❡♦♠❡tr✐❝ ❦♥♦✇❧❡❞❣❡ ❛r❡ ❡t❡r♥❛❧✧✳ ❚❤❡●r❡❡❦ ❞❡❞✉❝t✐✈❡ ♠❡t❤♦❞ ❣✐✈❡s ❛ ❦✐♥❞ ♦❢ ❛♥s✇❡r t♦ t❤❡ q✉❡st✐♦♥ ❤♦✇ t♦ ♦❜t❛✐♥✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤✐s ✐❞❡❛❧✐③❡❞ ✇♦r❧❞✳ ■t ✇❛s ❝♦❞✐✜❡❞ ❜② ❊✉❝❧✐❞ ❛r♦✉♥❞ ✸✵✵❇❈ ✐♥ ❤✐s ❢❛♠♦✉s ❜♦♦❦ ❡♥t✐t❧❡❞ ❊❧❡♠❡♥ts ✇❤✐❝❤ ✐s ❛ s②st❡♠ ♦❢ ❝♦♥❝❧✉s✐♦♥s♦♥ t❤❡ ❜❛s❡s ♦❢ ✉♥q✉❡st✐♦♥❛❜❧❡ ♣r❡♠✐ss❡s ♦r ❛①✐♦♠s✳ ■♥ t❡r♠s ♦❢ ❛ ♠♦❞✲❡r♥ ❧❛♥❣✉❛❣❡ t❤❡ ♠❡t❤♦❞ ♥❡❡❞s t✇♦ ❢✉♥❞❛♠❡♥t❛❧ ❝♦♥❝❡♣ts t♦ ❜❡❣✐♥ ✇♦r❦✐♥❣✿✉♥❞❡✜♥❡❞ t❡r♠s s✉❝❤ ❛s ♣♦✐♥ts✱ ❧✐♥❡s✱ ♣❧❛♥❡s ❡t❝✳ ❛♥❞ ❛①✐♦♠s ✭s♦♠❡t✐♠❡st❤❡② ❛r❡ r❡❢❡rr❡❞ ❛s ♣r❡♠✐ss❡s ♦r ♣♦st✉❧❛t❡s✮ ✇❤✐❝❤ ❛r❡ t❤❡ ❜❛s✐❝ ❛ss✉♠♣t✐♦♥s❛❜♦✉t t❤❡ t❡r♠s ♦❢ ❣❡♦♠❡tr②✳ ❍❡r❡ ✇❡ ♣r❡s❡♥t ❛ s❤♦rt r❡✈✐❡✇ ♦❢ ❛①✐♦♠s ✐♥ ❊✉✲❝❧✐❞❡❛♥ ♣❧❛♥❡ ❣❡♦♠❡tr② t♦ ✐❧❧✉str❛t❡ ✐ts ❢✉♥❞❛♠❡♥t❛❧ ❛ss✉♠♣t✐♦♥s✱ ♠❡t❤♦❞s❛♥❞ s♣❡❝✐✜❝ ♣♦✐♥ts ♦❢ ✈✐❡✇✳

✸✳✶ ❚❤❡ ❛①✐♦♠s ♦❢ ✐♥❝✐❞❡♥❝❡

❚❤❡ ❛①✐♦♠s ♦❢ ✐♥❝✐❞❡♥❝❡✳

• ❚❤r♦✉❣❤ ❛♥② t✇♦ ❞✐st✐♥❝t ♣♦✐♥ts t❤❡r❡ ✐s ❡①❛❝t❧② ♦♥❡ ❧✐♥❡✳

❚❤❡ ❜❛s✐❝ t❡r♠s ✭❧✐❦❡ ♣♦✐♥ts✱ ❧✐♥❡s ❡t❝✳✮ ♦❢ t❤❡ ❛①✐♦♠❛t✐❝ s②st❡♠ ❛r❡✉♥❞❡✜♥❡❞✳ ■❢ ✇❡ ❞♦ ♥♦t ❦♥♦✇ ✇❤❛t t❤❡② ♠❡❛♥ t❤❡♥ t❤❡r❡ ✐s ♥♦ ♣♦✐♥t ✐♥ ❛s❦✐♥❣✇❤❡t❤❡r ♦r ♥♦t t❤❡ ❛①✐♦♠s ❛r❡ tr✉❡✳ ❋♦❧❧♦✇✐♥❣ ♦♥❡ ♦❢ t❤❡ ♠♦st ❡①♣r❡ss✐✈❡❡①❛♠♣❧❡s ✐♥ ❬✶❪ s✉♣♣♦s❡ t❤❛t ❛❧✐❡♥ ❜❡✐♥❣s ❤❛✈❡ ❧❛♥❞❡❞ ♦♥ ❊❛rt❤ ❜② ✢②✐♥❣s❛✉❝❡r ❛♥❞ t❤❡✐r ❧❡❛❞❡r t❡❧❧s ②♦✉ t❤❛t t❤r♦✉❣❤ ❛♥② ❞✐st✐♥❝t ❜❧✉r❣s t❤❡r❡ ✐s❡①❛❝t❧② ♦♥❡ ♣❤♦❣♦♥✳ ❯♥❧❡ss ②♦✉ ❦♥♦✇ ✇❤❛t ❛ ❜❧✉r❣ ❛♥❞ ❛ ♣❤♦❣♦♥ ②♦✉ ✇✐❧❧❤❛✈❡ ♥♦ ✇❛② ♦❢ t❡❧❧✐♥❣ ✇❤❡t❤❡r ♦r ♥♦t t❤✐s st❛t❡♠❡♥t ✐s tr✉❡✳ ❖♥ t❤❡ ♦t❤❡r❤❛♥❞ t❤❡r❡ ♠❛② ❜❡ ♠❛♥② ❞✐✛❡r❡♥t ✐♥t❡r♣r❡t❛t✐♦♥s ♦❢ t❤❡ ✉♥❞❡✜♥❡❞ t❡r♠s s✉❝❤❛s ♣♦✐♥ts✱ ❧✐♥❡s ❡t❝✳ ✐♥ ❛♥ ❛①✐♦♠❛t✐❝ s②st❡♠ ❢♦r ❣❡♦♠❡tr②✳ ❆♥ ✐♥t❡r♣r❡t❛t✐♦♥✇❤✐❝❤ ♠❛❦❡s ❛❧❧ t❤❡ ❛①✐♦♠s tr✉❡ ✐s ❝❛❧❧❡❞ ❛ ♠♦❞❡❧ ❢♦r t❤❡ ❛①✐♦♠❛t✐❝ s②st❡♠❀

✺✼

✺✽ ❈❍❆P❚❊❘ ✸✳ ❇❆❙■❈ ❋❆❈❚❙ ■◆ ●❊❖▼❊❚❘❨

❜❡❝❛✉s❡ t❤❡♦r❡♠s ❛r❡ ❛❧❧ ❞❡❞✉❝❡❞ ❧♦❣✐❝❛❧❧② ❢r♦♠ t❤❡ ❛①✐♦♠s t❤❡② ✇✐❧❧ ❜❡ tr✉❡✐♥ ❛♥② ♠♦❞❡❧ ❛s ✇❡❧❧✳ ❚♦ ✉♥❞❡rst❛♥❞ t❤❡ r♦❧❡ ♦❢ ♠♦❞❡❧s ✇❡ ❝❛♥ ❝♦♥s✐❞❡r t❤❡❝❧❛ss✐❝❛❧ ❝♦♦r❞✐♥❛t❡ ❣❡♦♠❡tr② ❛s ♦♥❡ ♦❢ t❤❡ ♠♦❞❡❧ ❢♦r t❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡❣❡♦♠❡tr②✳ P♦✐♥ts ❛r❡ ✐♥t❡r♣r❡t❡❞ ❛s ♣❛✐rs ♦❢ r❡❛❧ ♥✉♠❜❡rs ✭❝♦♦r❞✐♥❛t❡s✮ ❛♥❞❧✐♥❡s ❛r❡ ✐♥t❡r♣r❡t❡❞ ❛s ♣♦✐♥t ✲ s❡ts s❛t✐s❢②✐♥❣ ❡q✉❛t✐♦♥s ♦❢ s♣❡❝✐❛❧ t②♣❡✳ ■♥t❤✐s ✐♥t❡r♣r❡t❛t✐♦♥ t❤❡ ✜rst ❛①✐♦♠ ♦❢ ✐♥❝✐❞❡♥❝❡ ❝❛♥ ❜❡ ❝❤❡❝❦❡❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣✇❛②✿ ❝♦♥s✐❞❡r t❤❡ ♣♦✐♥ts ✭x1✱ y1✮ ❛♥❞ ✭x2✱ y2✮ ✐♥ t❤❡ ♣❧❛♥❡❀ t❤❡ ❧✐♥❡ ♣❛ss✐♥❣t❤r♦✉❣❤ t❤❡s❡ ♣♦✐♥ts ✐s ❥✉st t❤❡ s❡t ♦❢ ♣♦✐♥ts ✇❤♦s❡ ❝♦♦r❞✐♥❛t❡s s❛t✐s❢② t❤❡❡q✉❛t✐♦♥

y − y1y2 − y1

=x− x1

x2 − x1

♣r♦✈✐❞❡❞ t❤❛t t❤❡ ✜rst ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ❣✐✈❡♥ ♣♦✐♥ts ❛r❡ ❞✐✛❡r❡♥t✳ ■♥ ❝❛s❡♦❢ x1 = x2 t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❧✐♥❡ ✐s ❥✉st ①❂ ❝♦♥st❛♥t✳

❘❡♠❛r❦ ❆s ✇❡ ❤❛✈❡ s❡❡♥ ❛❜♦✈❡ ♣♦✐♥ts ❝❛♥ ❜❡ ✐♥t❡r♣r❡t❡❞ ❛s ♣❛✐rs ♦❢ r❡❛❧♥✉♠❜❡rs✳ ❚❤❡ ❧✐♥❡s ❝♦rr❡s♣♦♥❞ t♦ ♠♦r❡ ❝♦♠♣❧✐❝❛t❡❞ ❛❧❣❡❜r❛✐❝ ♦❜❥❡❝ts ❝❛❧❧❡❞❡q✉❛t✐♦♥s✳ ❚❤✐s ✐s t❤❡ r❡❛s♦♥ ✇❤② s✉❝❤ ❛ ♠♦❞❡❧ ❢♦r t❤❡ ❊✉❝❧✐❞❡❛♥ ❣❡♦♠❡tr②✐s ❝❛❧❧❡❞ ❛♥❛❧②t✐❝✳ ■t ❝❛♥ ❜❡ ❡❛s✐❧② ❣❡♥❡r❛❧✐③❡❞ ❜② ❛❞♠✐tt✐♥❣ ♠♦r❡ t❤❛♥ t✇♦❝♦♦r❞✐♥❛t❡s✳ ❚❤✐s r❡s✉❧ts ✐♥ t❤❡ ❣❡♦♠❡tr② ♦❢ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥❛❧ ❊✉❝❧✐❞❡❛♥s♣❛❝❡s✳ ❚♦ ✐❧❧✉str❛t❡ ✇❤❛t ❤❛♣♣❡♥s ♥♦t❡ t❤❛t ❧✐♥❡s ✐♥ t❤❡ s♣❛❝❡ ❤❛✈❡ s②st❡♠♦❢ ❡q✉❛t✐♦♥s ♦❢ t❤❡ ❢♦r♠

z − z1z2 − z1

=y − y1y2 − y1

=x− x1

x2 − x1

♦rz = z1,

y − y1y2 − y1

=x− x1

x2 − x1

✐♥ ❝❛s❡ ♦❢ z1 = z2 ❛♥❞ s♦ ♦♥✳

• ❆♥② ❧✐♥❡ ❝♦♥t❛✐♥s ❛t ❧❡❛st t✇♦ ❞✐st✐♥❝t ♣♦✐♥ts ❛♥❞ ✇❡ ❤❛✈❡ ❛t ❧❡❛st t❤r❡❡❞✐st✐♥❝t ♣♦✐♥ts ✇❤✐❝❤ ❞♦ ♥♦t ❧✐❡ ♦♥ t❤❡ s❛♠❡ ❧✐♥❡✳

❚❤❡ st❛t❡♠❡♥t ✐s ❧❛❜❡❧❧❡❞ ❛s t❤❡ ❞✐♠❡♥s✐♦♥ ❛①✐♦♠ ❜❡❝❛✉s❡ ✐t s❛②s ❡ss❡♥✲t✐❛❧❧② t❤❛t ❧✐♥❡s ❛r❡ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ❛♥❞ t❤❡ ♣❧❛♥❡ ✐s ♦❢ ❞✐♠❡♥s✐♦♥ t✇♦✳

❉❡✜♥✐t✐♦♥ P♦✐♥ts ❧②✐♥❣ ♦♥ t❤❡ s❛♠❡ ❧✐♥❡ ❛r❡ ❝❛❧❧❡❞ ❝♦❧❧✐♥❡❛r✳

✸✳✷ P❛r❛❧❧❡❧✐s♠

❋✐♥❛❧❧② ✇❡ ♣r❡s❡♥t t❤❡ ♠♦st ❢❛♠♦✉s ❛①✐♦♠ ♦❢ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡ ❣❡♦♠❡tr② ✇❤✐❝❤❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ ✐♥❝✐❞❡♥❝❡✳ ❚❤✐s ✐s ❝❛❧❧❡❞ ❊✉❝❧✐❞✬s ♣❛r❛❧❧❡❧ ♣♦s✲t✉❧❛t❡✳

✸✳✸✳ ▼❊❆❙❯❘❊▼❊◆❚ ❆❳■❖▼❙ ✺✾

❉❡✜♥✐t✐♦♥ ❚✇♦ ❧✐♥❡s ✐♥ t❤❡ ♣❧❛♥❡ ❛r❡ ♣❛r❛❧❧❡❧ ✐❢ t❤❡② ❤❛✈❡ ♥♦ ❛♥② ♣♦✐♥t ✐♥❝♦♠♠♦♥ ♦r t❤❡② ❝♦✐♥❝✐❞❡✳

• ▲❡t ❧ ❜❡ ❛ ❧✐♥❡ ❛♥❞ P ❜❡ ❛ ♣♦✐♥t ✐♥ t❤❡ ♣❧❛♥❡❀ t❤❡r❡ ✐s ♦♥❡ ❛♥❞ ♦♥❧② ♦♥❡❧✐♥❡ ❡ t❤❛t ♣❛ss❡s t❤r♦✉❣❤ P ❛♥❞ ♣❛r❛❧❧❡❧ t♦ ❧✳

❚❤❡♦r❡♠ ✸✳✷✳✶ ■❢ ❧ ✐s ♣❛r❛❧❧❡❧ t♦ ❡ ❛♥❞ ❡ ✐s ♣❛r❛❧❧❡❧ t♦ ♠ t❤❡♥ ❧ ✐s ♣❛r❛❧❧❡❧t♦ ♠✳

Pr♦♦❢ ❙✉♣♣♦s❡ t❤❛t ❧ ❛♥❞ ♠ ❤❛s ❛ ♣♦✐♥t P ✐♥ ❝♦♠♠♦♥✳ ❙✐♥❝❡ ❜♦t❤ ♦❢ t❤❡❧✐♥❡s ❛r❡ ♣❛r❛❧❧❡❧ t♦ ❡ ✇❡ ❤❛✈❡ ❜② t❤❡ ♣❛r❛❧❧❡❧ ❛①✐♦♠ t❤❛t ❧❂♠✳ ❖t❤❡r✇✐s❡t❤❡② ❛r❡ ❞✐s❥♦✐♥t✳

❘❡♠❛r❦ ❉❡✜♥✐t✐♦♥s ❛r❡ s❤♦rt❝✉t ♥♦t❛t✐♦♥s ❢r♦♠ t❤❡ ❧♦❣✐❝❛❧ ♣♦✐♥t ♦❢ ✈✐❡✇✳❚❤❡♦r❡♠s ❛r❡ ❞❡❞✉❝❡❞ ❧♦❣✐❝❛❧❧② ❢r♦♠ t❤❡ ❛①✐♦♠s ♦r ♦t❤❡r t❤❡♦r❡♠s ✇❤✐❝❤❤❛s ❜❡❡♥ ♣r♦✈❡❞✳

✸✳✸ ▼❡❛s✉r❡♠❡♥t ❛①✐♦♠s

❆♥♦t❤❡r ✐♠♣♦rt❛♥t q✉❡st✐♦♥ ✐s ❤♦✇ t♦ ♠❡❛s✉r❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ♣♦✐♥ts ✐♥t❤❡ ♣❧❛♥❡✳ ▲✐❦❡ ♣♦✐♥ts✱ ❧✐♥❡s ❡t❝✳ t❤❡ ❛❜s♦❧✉t❡ ❞✐st❛♥❝❡ ❝❛♥ ❛❧s♦ ❜❡ ❛ ♥❡✇✉♥❞❡✜♥❡❞ t❡r♠ ✐♥ ♦✉r ❣❡♦♠❡tr②✳ ❚❤❡ ♠❛✐♥ q✉❡st✐♦♥ ✐s ♥♦t ✇❤❛t ✐s t❤❡ ❞✐s✲t❛♥❝❡ ❜✉t ❤♦✇ t♦ ♠❡❛s✉r❡ t❤❡ ❞✐st❛♥❝❡✳ ❚❤❡ ♣❤②s✐❝❛❧ ✐♥str✉♠❡♥t t♦ r❡❛❧✐③❡❞✐st❛♥❝❡ ♠❡❛s✉r❡♠❡♥ts ✐s ❛ r✉❧❡r✳ ■ts ❛❜str❛❝t ✭✐❞❡❛❧✐③❡❞✮ ✈❡rs✐♦♥ ✐s ❝❛❧❧❡❞t❤❡ r✉❧❡r ❛①✐♦♠✳

• ▲❡t ❧ ❜❡ ❛♥ ❛r❜✐tr❛r② ❧✐♥❡ ✐♥ t❤❡ ♣❧❛♥❡✳ ❆ r✉❧❡r ❢♦r ❧ ✐s ❛ ♦♥❡✲t♦✲♦♥❡❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ♣♦✐♥ts ✐♥ ❧ ❛♥❞ t❤❡ s❡t ♦❢ r❡❛❧ ♥✉♠❜❡rs✐♥ s✉❝❤ ❛ ✇❛② t❤❛t t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ♣♦✐♥ts ❆ ❛♥❞ ❇ ✐♥ ❧ ✐s❥✉st t❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ r❡❛❧s✿ ✐❢ ❆❝♦rr❡s♣♦♥❞s t♦ t❤❡ r❡❛❧ ♥✉♠❜❡r ❛ ❛♥❞ ❇ ❝♦rr❡s♣♦♥❞s t♦ t❤❡ r❡❛❧ ♥✉♠❜❡r❜ t❤❡♥

d(A,B) = |a− b|.❚❤❡ r✉❧❡r ❛①✐♦♠ ♣♦st✉❧❛t❡s t❤❡ ❡①✐st❡♥❝❡ ♦❢ s✉❝❤ ❛ r✉❧❡r ❢♦r ❛♥② ❧✐♥❡ ✐♥t❤❡ ♣❧❛♥❡✳

❇② t❤❡ ❤❡❧♣ ♦❢ ❛ r✉❧❡r ✇❡ ❝❛♥ ✉s❡ t❤❡ st❛♥❞❛r❞ ♦r❞❡r✐♥❣ ❛♠♦♥❣ r❡❛❧ ♥✉♠✲❜❡rs t♦ ❞❡✜♥❡ s❡❣♠❡♥ts ❛♥❞ ❤❛❧❢ ✲ ❧✐♥❡s✳ ▲❡t ❆ ❛♥❞ ❇ ❜❡ t✇♦ ❞✐st✐♥❝t ♣♦✐♥ts✐♥ t❤❡ ♣❧❛♥❡ ❛♥❞ ❝♦♥s✐❞❡r t❤❡ ❧✐♥❡ ❧ ♣❛ss✐♥❣ t❤r♦✉❣❤ t❤❡ ❣✐✈❡♥ ♣♦✐♥ts✳ ■❢ ❛ ❁❜ t❤❡♥ t❤❡ str❛✐❣❤t ❧✐♥❡ s❡❣♠❡♥t ❥♦✐♥✐♥❣ ❆ ❛♥❞ ❇ ✐s ❞❡✜♥❡❞ ❛s

AB := { C ∈ l | a ≤ c ≤ b},

✻✵ ❈❍❆P❚❊❘ ✸✳ ❇❆❙■❈ ❋❆❈❚❙ ■◆ ●❊❖▼❊❚❘❨

❋✐❣✉r❡ ✸✳✶✿ ❈♦♥❣r✉❡♥❝❡ ❛①✐♦♠✳

✇❤❡r❡ t❤❡ ♣♦✐♥ts ❝♦rr❡s♣♦♥❞ t♦ t❤❡ r❡❛❧ ♥✉♠❜❡rs ❛✱ ❜ ❛♥❞ ❝ ✉♥❞❡r ❛ r✉❧❡r✳❚❤❡ ❤❛❧❢ ✲ ❧✐♥❡ st❛rt✐♥❣ ❢r♦♠ ❆ t♦ ❇ ✐s ❝r❡❛t❡❞ ❜② ❝✉tt✐♥❣ t❤❡ ♣♦✐♥ts ✇✐t❤❝♦♦r❞✐♥❛t❡s ❝ ❁ ❛✳ ❙❡❣♠❡♥ts ❛♥❞ ❤❛❧❢ ✲ ❧✐♥❡s ❝♦rr❡s♣♦♥❞ t♦ ✐♥t❡r✈❛❧s ♦❢ t❤❡❢♦r♠ ❬❛✱❜❪ ✇❤❡r❡ t❤❡ st❛rt✐♥❣ ♦r t❤❡ ❡♥❞ ♣♦✐♥t ❝❛♥ ❜❡ ♣♦s✐t✐♦♥❡❞ ❛t ♣❧✉s ♦r♠✐♥✉s ✐♥✜♥✐t②✳

❚❤❡r❡ ❛r❡ s❡✈❡r❛❧ ✇❛②s ♦❢ ✐♥tr♦❞✉❝✐♥❣ t❤❡ ❝♦♥❝❡♣t ♦❢ ❛♥❣❧❡ ✐♥ ❣❡♦♠❡tr②✳❍❡r❡ ✇❡ ❝♦♥s✐❞❡r t❤✐s ❝♦♥❝❡♣t ❛s ❛ ♥❡✇ ✉♥❞❡✜♥❡❞ t❡r♠ ❣♦✈❡r♥❡❞ ❜② ✐ts ♦✇♥❛①✐♦♠s✳ ■♥st❡❛❞ ♦❢ t❤❡ ♣r❡❝✐s❡ ❢♦r♠✉❧❛t✐♦♥ ✇❡ ❛❝❝❡♣t t❤❛t t❤❡ ♣r♦tr❛❝t♦r❛①✐♦♠ ❢♦r♠✉❧❛t❡s t❤❡ ❛❜str❛❝t ✭✐❞❡❛❧✐③❡❞✮ ✈❡rs✐♦♥ ♦❢ t❤❡ ♣❤②s✐❝❛❧ ✐♥str✉♠❡♥t❢♦r ♠❡❛s✉r✐♥❣ ❛♥❣❧❡s ✐♥ t❤❡ r❡❛❧ ✇♦r❧❞✳

✸✳✹ ❈♦♥❣r✉❡♥❝❡ ❛①✐♦♠

❯s✐♥❣ ❛ r✉❧❡r ❛♥❞ ❛ ♣r♦tr❛❝t♦r ✇❡ ❝❛♥ ❝♦♠♣❛r❡ ❛♥❞ ❝♦♣② s❡❣♠❡♥ts ❛♥❞ ❛♥❣❧❡s✐♥ t❤❡ ♣❧❛♥❡✳ ❚❤❡ ♥❡①t ✐♠♣♦rt❛♥t q✉❡st✐♦♥ ✐s ❤♦✇ t♦ ❝♦♠♣❛r❡ ❛♥❞ ❝♦♣②tr✐❛♥❣❧❡s✳

▲❡t ❛ tr✐❛♥❣❧❡ ❆❇❈ ❜❡ ❣✐✈❡♥ ✐♥ t❤❡ ♣❧❛♥❡ ❛♥❞ ❝♦♥s✐❞❡r ❛♥ ❛r❜✐tr❛r② ❤❛❧❢ ✲❧✐♥❡ st❛rt✐♥❣ ❢r♦♠ ❛ ♣♦✐♥t ❆✬✳ ❯s✐♥❣ ❛ r✉❧❡r ✇❡ ❝❛♥ ❝♦♣② t❤❡ s❡❣♠❡♥t ❆❇ ❢r♦♠❆✬ ✐♥t♦ t❤❡ ❣✐✈❡♥ ❞✐r❡❝t✐♦♥✳ ❚❤✐s r❡s✉❧ts ✐♥ ❛ ♣♦✐♥t ❇✬ s✉❝❤ t❤❛t ❆❇❂❆✬❇✬✳❯s✐♥❣ ❛ ♣r♦tr❛❝t♦r ❛♥❞ ❛ r✉❧❡r ❛❣❛✐♥ ✇❡ ❝❛♥ ❝♦♥str✉❝t ❛ ♣♦✐♥t ❈✬ s✉❝❤ t❤❛t

α = t❤❡ ♠❡❛s✉r❡ ♦❢ 6 CAB = t❤❡ ♠❡❛s✉r❡ ♦❢ 6 C ′A′B′ = α′

❛♥❞ ❆✬❈✬❂❆❈✳ ❲❤❛t ❛❜♦✉t t❤❡ t❤❡ ♠✐ss✐♥❣ s✐❞❡s ❇❈ ❛♥❞ ❇✬❈✬✱ t❤❡ ♠✐ss✐♥❣❛♥❣❧❡s γ ❛♥❞ γ✬ ♦r β ❛♥❞ β✬❄ ❯♥❢♦rt✉♥❛t❡❧② ✇❡ ❝❛♥ ♥♦t ❦♥♦✇ ❛♥②t❤✐♥❣ ❛❜♦✉tt❤❡♠ ❜❡❝❛✉s❡ ♥♦r t❤❡ ❛①✐♦♠s ♦❢ ✐♥❝✐❞❡♥❝❡ ♥❡✐t❤❡r t❤❡ ♠❡❛s✉r❡♠❡♥t ❛①✐♦♠s

✸✳✺✳ ❆❘❊❆ ✻✶

❝❛rr② ❛♥② ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ ♠✐ss✐♥❣ ❞❛t❛ ♦❢ t❤❡ tr✐❛♥❣❧❡s✳ ■❢ ✇❡ ✇❛♥t t♦♠❛❦❡ t❤❡♠ ❝♦♥❣r✉❡♥t t❤❡♥ ✇❡ ❤❛✈❡ t♦ ♣♦st✉❧❛t❡ t❤❡♠ t♦ ❜❡ ❝♦♥❣r✉❡♥t✳

❉❡✜♥✐t✐♦♥ ■❢ t❤❡r❡ ✐s ❛ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ✈❡rt✐❝❡s ♦❢ t✇♦ tr✐❛♥❣❧❡s✐♥ s✉❝❤ ❛ ✇❛② t❤❛t ❛❧❧ ❝♦rr❡s♣♦♥❞✐♥❣ s✐❞❡s ❛♥❞ ❛❧❧ ❝♦rr❡s♣♦♥❞✐♥❣ ❛♥❣❧❡s ❛r❡❝♦♥❣r✉❡♥t t❤❡♥ t❤❡ tr✐❛♥❣❧❡s ❛r❡ ❝♦♥❣r✉❡♥t ❝♦♣② ♦❢ ❡❛❝❤ ♦t❤❡r✳

❚❤❡ ❝♦♥❣r✉❡♥❝❡ ❛①✐♦♠ ❛❧❧♦✇s ✉s t♦ ❞❡❞✉❝❡ t❤❡ ❝♦♥❣r✉❡♥❝❡ ♦❢ tr✐❛♥❣❧❡s✉♥❞❡r ❛ r❡❞✉❝❡❞ s②st❡♠ ♦❢ ✐♥❢♦r♠❛t✐♦♥✳

• ■❢ t❤❡r❡ ✐s ❛ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ✈❡rt✐❝❡s ♦❢ t✇♦ tr✐❛♥❣❧❡s ✐♥s✉❝❤ ❛ ✇❛② t❤❛t t✇♦ s✐❞❡s ❛♥❞ t❤❡ ❛♥❣❧❡ ❡♥❝❧♦s❡❞ ❜② t❤❡♠ ✐♥ ♦♥❡ ♦❢ t❤❡tr✐❛♥❣❧❡s ❛r❡ ❝♦♥❣r✉❡♥t t♦ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s✐❞❡s ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✲✐♥❣ ❛♥❣❧❡ ✐♥ t❤❡ s❡❝♦♥❞ ♦❢ t❤❡ tr✐❛♥❣❧❡s t❤❡♥ t❤❡ tr✐❛♥❣❧❡s ❛r❡ ❝♦♥❣r✉❡♥t❝♦♣② ♦❢ ❡❛❝❤ ♦t❤❡r✳

❙♦♠❡t✐♠❡s ✐t ✐s r❡❢❡rr❡❞ ❛s s✐❞❡ ✲ ❛♥❣❧❡ ✲ s✐❞❡ ✲ ❛①✐♦♠ ♦r ❙❆❙ ✲ ❛①✐♦♠✳

✸✳✺ ❆r❡❛

❋♦r♠❛❧❧② s♣❡❛❦✐♥❣ ❬✶❪ ❛r❡❛ ❝❛♥ ❜❡ ❝♦♥s✐❞❡r❡❞ ❛s ❛ ♥❡✇ ✉♥❞❡✜♥❡❞ t❡r♠ ✐♥t❤❡ ❛①✐♦♠❛t✐❝ s②st❡♠ ♦❢ ❣❡♦♠❡tr②✳ ❙♦♠❡ ♦❜✈✐♦✉s r❡q✉✐r❡♠❡♥ts ❝❛♥ ❜❡ ❢♦r✲♠✉❧❛t❡❞ ❛s ❢♦❧❧♦✇s✳ ▲❡t ❛ ♣♦❧②❣♦♥❛❧ r❡❣✐♦♥ ❜❡ ❞❡✜♥❡❞ ❛s t❤❡ ✜♥✐t❡ ✉♥✐♦♥ ♦❢tr✐❛♥❣❧❡s s✉❝❤ t❤❛t t❤❡ ♠❡♠❜❡rs ♦❢ t❤❡ ✉♥✐♦♥ ❤❛✈❡ ❛t ♠♦st ❝♦♠♠♦♥ s✐❞❡s♦r ✈❡rt✐❝❡s✳ ❚❤❡ ❛r❡❛ ♦❢ ❛ ❜♦✉♥❞❡❞ ♣♦❧②❣♦♥❛❧ r❡❣✐♦♥ ✐s ❛ ♥♦♥✲♥❡❣❛t✐✈❡ r❡❛❧♥✉♠❜❡r s❛t✐s❢②✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s✿

• ✭❛r❡❛ ✐♥✈❛r✐❛♥❝❡ ❛①✐♦♠✮ ❚❤❡ ❛r❡❛ ✐s ✐♥✈❛r✐❛♥t ✉♥❞❡r t❤❡ ✐s♦♠❡tr✐❡s✭❝❤❛♣t❡r ✶✷✮ ♦❢ t❤❡ ♣❧❛♥❡✳

• ✭❛r❡❛ ❛❞❞✐t✐♦♥ ❛①✐♦♠✮ ❚❤❡ ❛r❡❛ ♦❢ t❤❡ ✉♥✐♦♥ ♦❢ t✇♦ ♣♦❧✐❣♦♥❛❧ r❡❣✐♦♥s✐s ❥✉st t❤❡ s✉♠ ♦❢ t❤❡ ❛r❡❛s ♦❢ t❤❡ r❡❣✐♦♥s ♣r♦✈✐❞❡❞ t❤❛t t❤❡② ❤❛✈❡ ❛t♠♦st ❝♦♠♠♦♥ s✐❞❡s ♦r ✈❡rt✐❝❡s✳

• ✭❛r❡❛ ♥♦r♠❛❧✐③❛t✐♦♥ ❛①✐♦♠✮ ❚❤❡ ❛r❡❛ ♦❢ ❛ r❡❝t❛♥❣❧❡ ♦❢ s✐❞❡s ❛ ❛♥❞ ❜ ✐s❥✉st ❛·❜✳

✸✳✻ ❇❛s✐❝ ❢❛❝ts ✐♥ ❣❡♦♠❡tr②

■♥ ✇❤❛t ❢♦❧❧♦✇s ✇❡ s✉♠♠❛r✐③❡ s♦♠❡ ❢✉rt❤❡r ❢❛❝ts ✇❤✐❝❤ ✇✐❧❧ ❜❡ ❢r❡q✉❡♥t❧②✉s❡❞ ✐♥ t❤❡ ❢♦rt❤❝♦♠✐♥❣ ♠❛t❡r✐❛❧✳ ❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ❡♠♣❤❛s✐③❡ t❤❛t t❤❡② ❛r❡♥♦t ♥❡❝❡ss❛r✐❧② ❛①✐♦♠s ❜✉t ✇❡ ♦♠✐t t❤❡ ♣r♦♦❢s ❢♦r t❤❡ s❛❦❡ ♦❢ s✐♠♣❧✐❝✐t②✳

✻✷ ❈❍❆P❚❊❘ ✸✳ ❇❆❙■❈ ❋❆❈❚❙ ■◆ ●❊❖▼❊❚❘❨

❋✐❣✉r❡ ✸✳✷✿ ❚r✐❛♥❣❧❡ ✐♥❡q✉❛❧✐t✐❡s✳

✸✳✻✳✶ ❚r✐❛♥❣❧❡ ✐♥❡q✉❛❧✐t✐❡s

❚❤❡② ❛r❡ s♣❡❝✐❛❧ ❢♦r♠s ♦❢ t❤❡ ❜❛s✐❝ ♣r✐♥❝✐♣❧❡ ✐♥ ❣❡♦♠❡tr② s❛②✐♥❣ t❤❛t t❤❡s❤♦rt❡st ✇❛② ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ✐s t❤❡ str❛✐❣❤t ❧✐♥❡ s❡❣♠❡♥t✳ ❈♦♥s✐❞❡r ❛tr✐❛♥❣❧❡ ✇✐t❤ ✈❡rt✐❝❡s A✱ B ❛♥❞ C✳ ▲❡t ✉s ❞❡♥♦t❡ t❤❡ s✐❞❡s ♦♣♣♦s✐t❡ t♦ t❤❡❝♦rr❡s♣♦♥❞✐♥❣ ✈❡rt✐❝❡s ❜② a✱ b ❛♥❞ c✳ ❚❤❡♥

a+ b > c, c+ a > b ❛♥❞ b+ c > a.

❈♦r♦❧❧❛r② ✸✳✻✳✶ ❋♦r t❤❡ s✐❞❡s ❛✱ ❜ ❛♥❞ ❝ ♦❢ ❛ tr✐❛♥❣❧❡

|a− b| < c, |a− c| < b ❛♥❞ |b− c| < a.

❘❡♠❛r❦ ■❢ ❛ ❁ ❜ ❁ ❝ t❤❡♥ t❤❡ ❝♦r♦❧❧❛r② s❛②s t❤❛t

• t❤❡ ✐♥t❡r✈❛❧ ❬❛✱❜❪ ❝❛♥ ❜❡ ❝♦✈❡r❡❞ ❜② t❤❡ t❤✐r❞ s✐❞❡ ♦❢ t❤❡ tr✐❛♥❣❧❡✱

• t❤❡ ✐♥t❡r✈❛❧ ❬❛✱❝❪ ❝❛♥ ❜❡ ❝♦✈❡r❡❞ ❜② t❤❡ s❡❝♦♥❞ s✐❞❡ ♦❢ t❤❡ tr✐❛♥❣❧❡✱

• t❤❡ ✐♥t❡r✈❛❧ ❬❜✱❝❪ ❝❛♥ ❜❡ ❝♦✈❡r❡❞ ❜② t❤❡ ✜rst s✐❞❡ ♦❢ t❤❡ tr✐❛♥❣❧❡✳

✸✳✻✳✷ ❍♦✇ t♦ ❝♦♠♣❛r❡ tr✐❛♥❣❧❡s ■ ✲ ❝♦♥❣r✉❡♥❝❡

❚❤❡ ❜❛s✐❝ ❝❛s❡s ♦❢ ❝♦♥❣r✉❡♥❝❡ ♦❢ tr✐❛♥❣❧❡s ❛r❡

• ❙❆❙ ✭t✇♦ s✐❞❡s ❛♥❞ t❤❡ ❛♥❣❧❡ ❡♥❝❧♦s❡❞ ❜② t❤❡♠✮✱ ✐✳❡✳

a = a′, b = b′ ❛♥❞ γ = γ′

✭s❡❡ ❝♦♥❣r✉❡♥❝❡ ❛①✐♦♠✮✳

• ❆❙❆ ✭♦♥❡ s✐❞❡ ❛♥❞ t❤❡ ❛♥❣❧❡s ♦♥ t❤✐s s✐❞❡✮✱ ✐✳❡✳

c = c′, α = α′ ❛♥❞ β = β′

✸✳✻✳ ❇❆❙■❈ ❋❆❈❚❙ ■◆ ●❊❖▼❊❚❘❨ ✻✸

❋✐❣✉r❡ ✸✳✸✿ ❈♦♥❣r✉❡♥t tr✐❛♥❣❧❡s✳

❋✐❣✉r❡ ✸✳✹✿ ❚❤❡ ❝❛s❡ ❙s❆

• ❙❆❆ ✭♦♥❡ s✐❞❡ ❛♥❞ t✇♦ ❛♥❣❧❡s✮✱ ✐✳❡✳

c = c′, α = α′ ❛♥❞ γ = γ′

• ❙❙❙ ✭❛❧❧ s✐❞❡s✮✱ ✐✳❡✳

a = a′, b = b′ ❛♥❞ c = c′

• ❙s❆ ✭t✇♦ s✐❞❡s ❛♥❞ t❤❡ ❛♥❣❧❡ ♦♣♣♦s✐t❡ t♦ t❤❡ ❧❛r❣❡r ♦♥❡✮✱

a = a′, b = b′ ❛♥❞ α = α′ ♣r♦✈✐❞❡❞ t❤❛t a > b.

❚❤❡♦r❡♠ ✸✳✻✳✷ ✭❚❤❡ ❣❡♦♠❡tr✐❝ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ t❤❡ ♣❡r♣❡♥❞✐❝✉❧❛r ❜✐✲s❡❝t♦r✮ ❚❤❡ ♣❡r♣❡♥❞✐❝✉❧❛r ❜✐s❡❝t♦r ♦❢ ❛ s❡❣♠❡♥t ✐s t❤❡ ❧♦❝✉s ♦❢ ♣♦✐♥ts ✐♥ t❤❡♣❧❛♥❡ ❤❛✈✐♥❣ t❤❡ s❛♠❡ ❞✐st❛♥❝❡ ❢r♦♠ ❡❛❝❤ ♦❢ t❤❡ ❡♥❞♣♦✐♥ts✳

Pr♦♦❢ ▲❡t ❆❇ ❜❡ ❛ s❡❣♠❡♥t ✇✐t❤ ♠✐❞♣♦✐♥t ❋ ❛♥❞ ❝♦♥s✐❞❡r t❤❡ ❧✐♥❡ ❧ t❤r♦✉❣❤❋ ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t ❧ ✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ❧✐♥❡ ❆❇✳ ■❢ ❳ ✐s ❛ ♣♦✐♥t ✐♥ ❧ t❤❡♥t❤❡ tr✐❛♥❣❧❡s ❆❋❳ ❛♥❞ ❇❋❳ ❛r❡ ♦❜✈✐♦✉s❧② ❝♦♥❣r✉❡♥t t♦ ❡❛❝❤ ♦t❤❡r ❜❡❝❛✉s❡ ♦❢

✻✹ ❈❍❆P❚❊❘ ✸✳ ❇❆❙■❈ ❋❆❈❚❙ ■◆ ●❊❖▼❊❚❘❨

❋✐❣✉r❡ ✸✳✺✿ ❇✐s❡❝t♦rs✳

t❤❡ ❝♦♥❣r✉❡♥❝❡ ❛①✐♦♠ ❙❆❙✳ ❚❤❡r❡❢♦r❡ ❆❳❂❇❳✳ ❈♦♥✈❡rs❡❧② ✐❢ ❆❳❂❇❳ t❤❡♥t❤❡ tr✐❛♥❣❧❡s ❆❋❳ ❛♥❞ ❇❋❳ ❛r❡ ❝♦♥❣r✉❡♥t ❜❡❝❛✉s❡ ♦❢ ❙❙❙✳ ❚❤❡r❡❢♦r❡ t❤❡❛♥❣❧❡s ❛t ❋ ❛r❡ ❡q✉❛❧ ❛♥❞ t❤❡✐r s✉♠ ✐s ✶✽✵ ❞❡❣r❡❡ ✐♥ ♠❡❛s✉r❡✳ ❚❤✐s ♠❡❛♥st❤❛t t❤❡ ❧✐♥❡ ❳❋ ✐s ❥✉st t❤❡ ♣❡r♣❡♥❞✐❝✉❧❛r ❜✐s❡❝t♦r ♦❢ t❤❡ s❡❣♠❡♥t✳

❊①❝❡r❝✐s❡ ✸✳✻✳✸ ❋♦r♠✉❧❛t❡ t❤❡ ❣❡♦♠❡tr✐❝ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ t❤❡ ❜✐s❡❝t♦r ♦❢❛♥ ❛♥❣❧❡ ✐♥ t❤❡ ♣❧❛♥❡✳

❍✐♥t✳ ❙✐♥❝❡ t❤❡ tr✐❛♥❣❧❡s ❋❳❆ ❛♥❞ ❋❳❇ ❛r❡ ❝♦♥❣r✉❡♥t t❤❡ ❜✐s❡❝t♦r ✐s t❤❡❧♦❝✉s ♦❢ ♣♦✐♥ts ✐♥ t❤❡ ♣❧❛♥❡ ❤❛✈✐♥❣ t❤❡ s❛♠❡ ❞✐st❛♥❝❡ ❢r♦♠ ❡❛❝❤ ♦❢ t❤❡ ❛r♠s♦❢ t❤❡ ❛♥❣❧❡✳

❚❤❡♦r❡♠ ✸✳✻✳✹ ❚❤❡ ♦r❞❡r✐♥❣ ❛♠♦♥❣ t❤❡ s✐❞❡s ♦❢ ❛ tr✐❛♥❣❧❡ ✐s t❤❡ s❛♠❡ ❛st❤❡ ♦r❞❡r✐♥❣ ❛♠♦♥❣ t❤❡ ❛♥❣❧❡s ♦❢ t❤❡ tr✐❛♥❣❧❡✳

✸✳✻✳✸ ❈❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ♣❛r❛❧❧❡❧✐s♠

❚❤❡ ❡ss❡♥t✐❛❧ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ♣❛r❛❧❧❡❧ ❛①✐♦♠ ❛♥❞ t❤❡ ♦t❤❡r ♦♥❡s ✐s❤✐❞❞❡♥ ✐♥ t❤❡ ♥♦t✐♦♥ ♦❢ ♣❛r❛❧❧❡❧✐s♠ ✐ts❡❧❢✳ ❚❤❡ ♣❛r❛❧❧❡❧✐s♠ ✐♥✈♦❧✈❡s t❤❡ ✐❞❡❛ ♦❢✐♥✜♥✐t② ✐♥ ❛ r❛t❤❡r ✐♠♣♦rt❛♥t ✇❛②✳ ■❢ ✇❡ ❦♥♦✇ t❤❛t t✇♦ ❧✐♥❡s ❛r❡ ♥♦t ♣❛r❛❧❧❡❧✇❡ st✐❧❧ ❤❛✈❡ ♥♦ ✐❞❡❛ ❤♦✇ ❢❛r ♦♥❡ ♠❛② ❤❛✈❡ t♦ tr❛❝❡ ❛❧♦♥❣ t❤❡♠ ❜❡❢♦r❡ t❤❡②❛❝t✉❛❧❧② ♠❡❡t✳ ❚❤❡ ✐❞❡❛ ♦❢ ✐♥✜♥✐t② ✐s ❛❧✇❛②s ♣r♦❜❧❡♠❛t✐❝ ❜❡❝❛✉s❡ ♠❛♥② ❡rr♦rs✐♥ ♠❛t❤❡♠❛t✐❝s ❛r✐s❡ ❢r♦♠ ❣❡♥❡r❛❧✐③❛t✐♦♥s t♦ t❤❡ ✐♥✜♥✐t❡ ♦❢ ✇❤❛t ✐s ❦♥♦✇♥tr✉❡ ❢♦r t❤❡ ✜♥✐t❡✳ ❆s ♦♥❡ ♦❢ ✐♥t❡r❡st✐♥❣ ❡①❛♠♣❧❡s ❝♦♥s✐❞❡r ❛ ❤♦t❡❧ ❤❛✈✐♥❣❛s ♠❛♥② r♦♦♠s ❛s ♠❛♥② ♥❛t✉r❛❧ ♥✉♠❜❡rs ✇❡ ❤❛✈❡✳ ■s ✐t ♣♦ss✐❜❧❡ t♦ ♣r♦✈✐❞❡❛❝❝♦♠♠♦❞❛t✐♦♥ ❢♦r ♦♥❡ ♠♦r❡ ❣✉❡st ✐❢ ❛❧❧ ♦❢ r♦♦♠s ❛r❡ ♦❝❝✉♣✐❡❞❄ ❚❤❡ ❛♥s✇❡r✐s ❞❡✜♥✐t❡❧② ②❡s ❜❡❝❛✉s❡ ✐❢ t❤❡ ❣✉❡st ✐♥ r♦♦♠ ♥ ✐s ♠♦✈✐♥❣ ✐♥t♦ r♦♦♠ ♥✰✶ t❤❡♥r♦♦♠ ✶ ❜❡❝♦♠❡s ❢r❡❡✳ ■♥ ✇❤❛t ❢♦❧❧♦✇s ✇❡ ♣r❡s❡♥t ❛ ♠❡t❤♦❞ ♦❢ ❝❤❡❝❦✐♥❣ t❤❡♣❛r❛❧❧❡❧✐s♠ ❜② ♠❡❛s✉r✐♥❣ ❛♥❣❧❡s ✐♥st❡❛❞ ♦❢ t❛❦✐♥❣ ❛♥ ✐♥✜♥✐t❡ ✲ ❧♦♥❣ ✇❛❧❦✳

❊①❝❡r❝✐s❡ ✸✳✻✳✺ ▲❡t e ❛♥❞ f ❜❡ ♣❛r❛❧❧❡❧ ❧✐♥❡s ✐♥ t❤❡ ♣❧❛♥❡ ❛♥❞ ❝♦♥s✐❞❡r ❛tr❛♥s✈❡rs❛❧ f ✳ ❋✐♥❞ t❤❡ r❡❧❛t✐♦♥s❤✐♣s ❛♠♦♥❣ t❤❡ ✐♥❝❧✐♥❛t✐♦♥ ❛♥❣❧❡s✳

✸✳✻✳ ❇❆❙■❈ ❋❆❈❚❙ ■◆ ●❊❖▼❊❚❘❨ ✻✺

❋✐❣✉r❡ ✸✳✻✿ ❈❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ♣❛r❛❧❧❡❧✐s♠✳

❚❤❡♦r❡♠ ✸✳✻✳✻ ✭❈❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ♣❛r❛❧❧❡❧✐s♠✮ ❚❤❡ ❧✐♥❡s e ❛♥❞ f ❛r❡ ♣❛r✲❛❧❧❡❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥s❤✐♣s ✐s tr✉❡ ❢♦r t❤❡ ✐♥❝❧✐♥❛t✐♦♥❛♥❣❧❡s✿

β = δ′, β + γ = 180◦ ♦r β = δ.

❊①❝❡r❝✐s❡ ✸✳✻✳✼ Pr♦✈❡ t❤❛t t❤❡ s✉♠ ♦❢ t❤❡ ✐♥t❡r✐♦r ❛♥❣❧❡s ♦❢ ❛ tr✐❛♥❣❧❡ ✐s✶✽✵ ❞❡❣r❡❡ ✐♥ ♠❡❛s✉r❡✳

❍✐♥t✳ ▲❡t ❆❇❈ ❜❡ ❛ tr✐❛♥❣❧❡✳ ❚❛❦✐♥❣ t❤❡ ❧✐♥❡ ❧ t❤r♦✉❣❤ t❤❡ ♣♦✐♥t ❈ ✐♥ s✉❝❤❛ ✇❛② t❤❛t ❧ ✐s ♣❛r❛❧❧❡❧ t♦ t❤❡ s✐❞❡ ❆❇✱ t❤❡ st❛t❡♠❡♥t ✐s ❛ ❞✐r❡❝t ❝♦♥s❡q✉❡♥❝❡♦❢ t❤❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ♣❛r❛❧❧❡❧✐s♠✳

✸✳✻✳✹ ❍♦✇ t♦ ❝♦♠♣❛r❡ tr✐❛♥❣❧❡s ■■ ✲ s✐♠✐❧❛r✐t②

❚❤❡♦r❡♠ ✸✳✻✳✽ ✭P❛r❛❧❧❡❧ ❧✐♥❡s ✐♥t❡rs❡❝t✐♥❣ t❤❡♦r❡♠✮ ▲❡t ❡ ❛♥❞ ❡✬ ❜❡ t✇♦❧✐♥❡s ✐♥ t❤❡ ♣❧❛♥❡ ♠❡❡t✐♥❣ ❛t t❤❡ ♣♦✐♥t ❖✳ ■❢ t❤❡ ❧✐♥❡s ❛ ❛♥❞ ❜ ❛r❡ ♣❛r❛❧❧❡❧ t♦❡❛❝❤ ♦t❤❡r s✉❝❤ t❤❛t t❤❡ ❧✐♥❡ ❛ ♠❡❡ts ❡ ❛♥❞ ❡✬ ❛t t❤❡ ♣♦✐♥ts ❆ ❛♥❞ ❆✬✱ t❤❡❧✐♥❡ ❜ ♠❡❡ts ❡ ❛♥❞ ❡✬ ❛t t❤❡ ♣♦✐♥ts ❇ ❛♥❞ ❇✬ t❤❡♥

OA : OB = OA′ : OB′.

■♥ t❤❡ ❍✉♥❣❛r✐❛♥ ❡❞✉❝❛t✐♦♥❛❧ tr❛❞✐t✐♦♥ ✐t ✐s ❛ t❤❡♦r❡♠✳ ■t ✐s ❛❧s♦ ♣♦ss✐❜❧❡t♦ ❝♦♥s✐❞❡r t❤❡ st❛t❡♠❡♥t ❛s ❛♥ ❛①✐♦♠❀ s❡❡ ❙✐♠✐❧❛r✐t② ❛①✐♦♠ ✐♥ ❬✶❪✳

❉❡✜♥✐t✐♦♥ ▲❡t ❡ ❛♥❞ ❡✬ ❜❡ t✇♦ ❧✐♥❡s ✐♥ t❤❡ ♣❧❛♥❡ ♠❡❡t✐♥❣ ❛t t❤❡ ♣♦✐♥t ❖✳❲❡ s❛② t❤❛t t❤❡ ♣♦✐♥ts ❖✱ ❆✱ ❇ ♦♥ t❤❡ ❧✐♥❡ ❡ ❝♦rr❡s♣♦♥❞ t♦ t❤❡ ♣♦✐♥ts ❖✱ ❆✬✱❇✬ ♦♥ t❤❡ ❧✐♥❡ ❡✬ ✐❢ t❤❡② ❤❛✈❡ t❤❡ s❛♠❡ ♦r❞❡r✐♥❣✱ ✐✳❡✳ t❤❡ ❧✐♥❡ ❡ s❡♣❛r❛t❡s ❆✬❛♥❞ ❇✬ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❧✐♥❡ ❡✬ s❡♣❛r❛t❡s ❆ ❛♥❞ ❇✳

✻✻ ❈❍❆P❚❊❘ ✸✳ ❇❆❙■❈ ❋❆❈❚❙ ■◆ ●❊❖▼❊❚❘❨

❋✐❣✉r❡ ✸✳✼✿ P❛r❛❧❧❡❧ ❧✐♥❡s ✐♥t❡rs❡❝t✐♥❣ t❤❡♦r❡♠✳

❋✐❣✉r❡ ✸✳✽✿ ❙✐♠✐❧❛r tr✐❛♥❣❧❡s✳

❚❤❡♦r❡♠ ✸✳✻✳✾ ✭❚❤❡ ❝♦♥✈❡rs❡ ♦❢ t❤❡ ♣❛r❛❧❧❡❧ ❧✐♥❡s ✐♥t❡rs❡❝t✐♥❣ t❤❡♦r❡♠✮ ▲❡t❡ ❛♥❞ ❡✬ ❜❡ t✇♦ ❧✐♥❡s ✐♥ t❤❡ ♣❧❛♥❡ ♠❡❡t✐♥❣ ❛t t❤❡ ♣♦✐♥t ❖✳ ■❢ t❤❡ ❧✐♥❡ ❛ ♠❡❡ts❡ ❛♥❞ ❡✬ ❛t t❤❡ ♣♦✐♥ts ❆ ❛♥❞ ❆✬✱ t❤❡ ❧✐♥❡ ❜ ♠❡❡ts ❡ ❛♥❞ ❡✬ ❛t t❤❡ ♣♦✐♥ts ❇ ❛♥❞❇✬ s✉❝❤ t❤❛t ❖✱ ❆✱ ❇ ❝♦rr❡s♣♦♥❞ t♦ ❖✱ ❆✬✱ ❇✬ ❛♥❞ OA : OB = OA′ : OB′

t❤❡♥ t❤❡ ❧✐♥❡s ❛ ❛♥❞ ❜ ❛r❡ ♣❛r❛❧❧❡❧✳

❚❤❡ ♣❛r❛❧❧❡❧ ❧✐♥❡s ✐♥t❡rs❡❝t✐♥❣ t❤❡♦r❡♠ ✭❛♥❞ ✐ts ❝♦♥✈❡rs❡✮ t♦❣❡t❤❡r ✇✐t❤t❤❡ ❜❛s✐❝ ❝❛s❡s ♦❢ t❤❡ ❝♦♥❣r✉❡♥❝❡ ♦❢ tr✐❛♥❣❧❡s ❣✐✈❡ ❛✉t♦♠❛t✐❝❛❧❧② t❤❡ ❜❛s✐❝❝❛s❡s ♦❢ s✐♠✐❧❛r✐t②✳

❉❡✜♥✐t✐♦♥ ■❢ t❤❡r❡ ✐s ❛ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ✈❡rt✐❝❡s ♦❢ t✇♦ tr✐❛♥❣❧❡s✐♥ s✉❝❤ ❛ ✇❛② t❤❛t ❛❧❧ ❝♦rr❡s♣♦♥❞✐♥❣ ❛♥❣❧❡s ❛r❡ ❝♦♥❣r✉❡♥t ❛♥❞ t❤❡ r❛t✐♦s❜❡t✇❡❡♥ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s✐❞❡s ❛r❡ ❛❧s♦ ❡q✉❛❧ t❤❡♥ t❤❡ tr✐❛♥❣❧❡s ❛r❡ s❛✐❞ t♦❜❡ s✐♠✐❧❛r✳

❚❤❡ ❜❛s✐❝ ❝❛s❡s ♦❢ s✐♠✐❧❛r✐t② ♦❢ tr✐❛♥❣❧❡s ❛r❡

• ❙✬❆❙✬ ✭t✇♦ s✐❞❡s ❛♥❞ t❤❡ ❛♥❣❧❡ ❡♥❝❧♦s❡❞ ❜② t❤❡♠✮✱ ✐✳❡✳

a : a = b : b′ ❛♥❞ γ = γ′.

• ❆❆❆ ✭❛❧❧ ♦❢ ❛♥❣❧❡s✮✱ ✐✳❡✳

α = α′, β = β′ ❛♥❞ γ = γ′,

✸✳✻✳ ❇❆❙■❈ ❋❆❈❚❙ ■◆ ●❊❖▼❊❚❘❨ ✻✼

• ❙✬❙✬❙✬ ✭❛❧❧ s✐❞❡s✮✱ ✐✳❡✳a : a′ = b : b′ = c : c′

• ❙✬s✬❆ ✭t✇♦ s✐❞❡s ❛♥❞ t❤❡ ❛♥❣❧❡ ♦♣♣♦s✐t❡ t♦ t❤❡ ❧❛r❣❡r ♦♥❡✮✱

a : a′ = b : b′ ❛♥❞ α = α′ ♣r♦✈✐❞❡❞ t❤❛t a > b.

✻✽ ❈❍❆P❚❊❘ ✸✳ ❇❆❙■❈ ❋❆❈❚❙ ■◆ ●❊❖▼❊❚❘❨

❈❤❛♣t❡r ✹

❚r✐❛♥❣❧❡s

✹✳✶ ●❡♥❡r❛❧ tr✐❛♥❣❧❡s ■

▲❡t ✉s st❛rt ✇✐t❤ t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❞✐st✐♥❣✉✐s❤❡❞ ♣♦✐♥ts✱ ❧✐♥❡s ❛♥❞ ❝✐r❝❧❡s r❡✲❧❛t❡❞ t♦ ❛ tr✐❛♥❣❧❡

❉❡✜♥✐t✐♦♥ ❚❤❡ ❧✐♥❡s ♣❛ss✐♥❣ t❤r♦✉❣❤ t❤❡ ♠✐❞♣♦✐♥ts ♦❢ t❤❡ s✐❞❡s ♦❢ ❛ tr✐❛♥❣❧❡❛r❡ ❝❛❧❧❡❞ ♠✐❞❧✐♥❡s✳

❯s✐♥❣ t❤❡ ❝♦♥✈❡rs❡ ♦❢ t❤❡ ♣❛r❛❧❧❡❧ ❧✐♥❡s ✐♥t❡rs❡❝t✐♥❣ t❤❡♦r❡♠ ✸✳✻✳✾ ✐t ❝❛♥❜❡ ❡❛s✐❧② s❡❡♥ t❤❛t ❛♥② ♠✐❞❧✐♥❡ ✐s ♣❛r❛❧❧❡❧ t♦ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s✐❞❡ ❛♥❞ t❤❡❧✐♥❡ s❡❣♠❡♥t ❜❡t✇❡❡♥ t❤❡ ♠✐❞♣♦✐♥ts ✐s ❥✉st t❤❡ ❤❛❧❢ ♦❢ t❤✐s s✐❞❡✳

❚❤❡♦r❡♠ ✹✳✶✳✶ ❚❤❡ ♣❡r♣❡♥❞✐❝✉❧❛r ❜✐s❡❝t♦rs ♦❢ t❤❡ s✐❞❡s ♦❢ ❛ tr✐❛♥❣❧❡ ❛r❡❝♦♥❝✉rr❡♥t ❛t ❛ ♣♦✐♥t ✇❤✐❝❤ ✐s ❥✉st t❤❡ ❝❡♥t❡r ♦❢ t❤❡ ❝✐r❝✉♠s❝r✐❜❡❞ ❝✐r❝❧❡✳

Pr♦♦❢ ❚❤❡ st❛t❡♠❡♥t ✐s ❛ ❞✐r❡❝t ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ ❣❡♦♠❡tr✐❝ ❝❤❛r❛❝t❡r✐③❛✲t✐♦♥ ✸✳✻✳✷ ♦❢ t❤❡ ❜✐s❡❝t♦r ♦❢ ❛ s❡❣♠❡♥t✳

❋✐❣✉r❡ ✹✳✶✿ ▼✐❞❧✐♥❡s

✻✾

✼✵ ❈❍❆P❚❊❘ ✹✳ ❚❘■❆◆●▲❊❙

❋✐❣✉r❡ ✹✳✷✿ ❈✐r❝✉♠❝✐r❝❧❡

❋✐❣✉r❡ ✹✳✸✿ ■♥❝✐r❝❧❡

❚❤❡♦r❡♠ ✹✳✶✳✷ ❚❤❡ ❜✐s❡❝t♦rs ♦❢ t❤❡ ✐♥t❡r✐♦r ❛♥❣❧❡s ♦❢ ❛ tr✐❛♥❣❧❡ ❛r❡ ❝♦♥✲❝✉rr❡♥t ❛t ❛ ♣♦✐♥t ✇❤✐❝❤ ✐s ❥✉st t❤❡ ❝❡♥t❡r ♦❢ t❤❡ ✐♥s❝r✐❜❡❞ ❝✐r❝❧❡✳

Pr♦♦❢ ❚❤❡ st❛t❡♠❡♥t ✐s ❛ ❞✐r❡❝t ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ ❣❡♦♠❡tr✐❝ ❝❤❛r❛❝t❡r✐③❛✲t✐♦♥ ♦❢ t❤❡ ❜✐s❡❝t♦r ♦❢ ❛♥ ❛♥❣❧❡✳

❚❤❡♦r❡♠ ✹✳✶✳✸ ❚❤❡ ❛❧t✐t✉❞❡s ♦❢ ❛ tr✐❛♥❣❧❡ ❛r❡ ❝♦♥❝✉rr❡♥t ❛t ❛ ♣♦✐♥t ✇❤✐❝❤✐s ❝❛❧❧❡❞ t❤❡ ♦rt❤♦❝❡♥t❡r ♦❢ t❤❡ tr✐❛♥❣❧❡✳

Pr♦♦❢ ❈♦♥s✐❞❡r t❤❡ tr✐❛♥❣❧❡ ❝♦♥st✐t✉t❡❞ ❜② t❤❡ ♣❛r❛❧❧❡❧ ❧✐♥❡s t♦ t❤❡ s✐❞❡s♣❛ss✐♥❣ t❤r♦✉❣❤ t❤❡ ♦♣♣♦s✐t❡ ✈❡rt✐❝❡s✳ ❚❤❡ ♦rt❤♦❝❡♥t❡r ♦❢ t❤❡ tr✐❛♥❣❧❡ ❆❇❈✐s t❤❡ ❝❡♥t❡r ♦❢ t❤❡ ❝✐r❝✉♠s❝r✐❜❡❞ ❝✐r❝❧❡ ♦❢ ❆✬❇✬❈✬✳

❉❡✜♥✐t✐♦♥ ❚❤❡ ♠❡❞✐❛♥s ♦❢ ❛ tr✐❛♥❣❧❡ ❛r❡ t❤❡ str❛✐❣❤t ❧✐♥❡s ❥♦✐♥✐♥❣ t❤❡ ✈❡r✲t✐❝❡s ❛♥❞ t❤❡ ♠✐❞♣♦✐♥ts ♦❢ t❤❡ ♦♣♣♦s✐t❡ s✐❞❡s✳

✹✳✷✳ ❚❍❊ ❊❯▲❊❘ ▲■◆❊ ❆◆❉ ❚❍❊ ❋❊❯❊❘❇❆❈❍ ❈■❘❈▲❊ ✼✶

❋✐❣✉r❡ ✹✳✹✿ ❖rt❤♦❝❡♥t❡r ✲ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡ ❛❧t✐t✉❞❡s

❋✐❣✉r❡ ✹✳✺✿ ❇❛r②❝❡♥t❡r ✲ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡ ♠❡❞✐❛♥s

❚❤❡♦r❡♠ ✹✳✶✳✹ ❚❤❡ ♠❡❞✐❛♥s ❛r❡ ❝♦♥❝✉rr❡♥t ❛t ❛ ♣♦✐♥t ✇❤✐❝❤ ✐s ❝❛❧❧❡❞ t❤❡❜❛r②❝❡♥t❡r✴❝❡♥tr♦✐❞ ♦❢ t❤❡ tr✐❛♥❣❧❡✳ ❚❤✐s ♣♦✐♥t ❞✐✈✐❞❡s t❤❡ ♠❡❞✐❛♥s ✐♥ t❤❡r❛t✐♦ 2 : 1✳

Pr♦♦❢ ■t ❝❛♥ ❡❛s✐❧② s❡❡♥ t❤❛t

• t❤❡ tr✐❛♥❣❧❡ ❋❙❉ ✐s s✐♠✐❧❛r t♦ t❤❡ tr✐❛♥❣❧❡ ❈❙❆✱

• t❤❡ tr✐❛♥❣❧❡ ❉❙❊ ✐s s✐♠✐❧❛r t♦ t❤❡ tr✐❛♥❣❧❡ ❆❙❇✳

❚❤❡ r❛t✐♦ ♦❢ t❤❡ s✐♠✐❧❛r✐t② ✐s ✶ ✿ ✷✳ ❚❤✐s ♠❡❛♥s t❤❛t t❤❡ ♠❡❞✐❛♥s ❇❊ ❛♥❞ ❈❋✐♥t❡rs❡❝t ❆❉ ✉♥❞❡r t❤❡ s❛♠❡ r❛t✐♦✳ ❚❤❡r❡❢♦r❡ t❤❡② ❛r❡ ❝♦♥❝✉rr❡♥t ❛t ❙✳

❘❡♠❛r❦ ❊❛❝❤ ♠❡❞✐❛♥ ❜✐s❡❝ts t❤❡ ❛r❡❛ ♦❢ t❤❡ tr✐❛♥❣❧❡✳

✹✳✷ ❚❤❡ ❊✉❧❡r ❧✐♥❡ ❛♥❞ t❤❡ ❋❡✉❡r❜❛❝❤ ❝✐r❝❧❡

❚❤❡♦r❡♠ ✹✳✷✳✶ ❚❤❡ ♦rt❤♦❝❡♥t❡r ▼✱ t❤❡ ❝❡♥t❡r ❖ ♦❢ t❤❡ ❝✐r❝✉♠s❝r✐❜❡❞ ❝✐r❝❧❡❛♥❞ t❤❡ ❜❛r②❝❡♥t❡r ❙ ❛r❡ ❝♦❧❧✐♥❡❛r✳ ❚❤❡ ♣♦✐♥t ❙ ❞✐✈✐❞❡s t❤❡ s❡❣♠❡♥t ▼❖ ✐♥t❤❡ r❛t✐♦ ✷ ✿ ✶✳ ❚❤❡ ❝♦♠♠♦♥ ❧✐♥❡ ♦❢ t❤❡ ♣♦✐♥ts ▼✱ ❖ ❛♥❞ ❙ ✐s ❝❛❧❧❡❞ t❤❡ ❊✉❧❡r❧✐♥❡✳

✼✷ ❈❍❆P❚❊❘ ✹✳ ❚❘■❆◆●▲❊❙

❋✐❣✉r❡ ✹✳✻✿ ❊✉❧❡r✲❧✐♥❡

❋✐❣✉r❡ ✹✳✼✿ ❈❡♥tr❛❧ s✐♠✐❧❛r✐t②

Pr♦♦❢ ❚❤❡ ♣r♦♦❢ ✐s ❜❛s❡❞ ♦♥ t❤❡ ❝❡♥tr❛❧ s✐♠✐❧❛r✐t② ✇✐t❤ r❡s♣❡❝t t♦ t❤❡❜❛r②❝❡♥t❡r✳ ❆ ❝❡♥tr❛❧ s✐♠✐❧❛r✐t② ✐s ❛ ♣♦✐♥t tr❛♥s❢♦r♠❛t✐♦♥ P → P ′ ♦❢ t❤❡♣❧❛♥❡ s✉❝❤ t❤❛t

• t❤❡r❡ ✐s ❛ ❞✐st✐♥❣✉✐s❤❡❞ ♣♦✐♥t ❈ ✇❤✐❝❤ ✐s t❤❡ ♦♥❧② ✜①♣♦✐♥t ✭❝❡♥t❡r✮ ♦❢t❤❡ tr❛♥s❢♦r♠❛t✐♦♥✱

• P✱ ❈ ❛♥❞ P✬ ❛r❡ ❝♦❧❧✐♥❡❛r✱

• t❤❡r❡ ✐s ❛ r❡❛❧ ♥✉♠❜❡r λ 6= 0 s✉❝❤ t❤❛t

CP ′ : CP = |λ|.

■❢ λ > 0 t❤❡♥ P ❛♥❞ P✬ ❛r❡ ♦♥ t❤❡ s❛♠❡ r❛② ❡♠❛♥❛t✐♥❣ ❢r♦♠ ❈✳ ■♥ ❝❛s❡♦❢ λ < 0 t❤❡ ❝❡♥t❡r s❡♣❛r❛t❡s P ❛♥❞ P✬✳

❆❝❝♦r❞✐♥❣ t♦ t❤❡ ❝♦♥✈❡rs❡ ♦❢ t❤❡ ♣❛r❛❧❧❡❧ ❧✐♥❡s ✐♥t❡rs❡❝t✐♥❣ t❤❡♦r❡♠ ❛♥②❧✐♥❡ ✐s ♣❛r❛❧❧❡❧ t♦ t❤❡ ✐♠❛❣❡ ✉♥❞❡r ❛ ❝❡♥tr❛❧ s✐♠✐❧❛r✐t②✳ ❈♦♥s✐❞❡r ♥♦✇ t❤❡ ❝❡♥✲tr❛❧ s✐♠✐❧❛r✐t② ✇✐t❤ ❝❡♥t❡r ❙ ❛♥❞ r❛t✐♦ ✲ ✶✴✷✳ ❚❤❡♥ ❡❛❝❤ ✈❡rt❡① ✐s tr❛♥s❢❡rr❡❞✐♥t♦ t❤❡ ♠✐❞♣♦✐♥t ♦❢ t❤❡ ♦♣♣♦s✐t❡ s✐❞❡ ❛♥❞ ❡❛❝❤ ❛❧t✐t✉❞❡ ✐s tr❛♥s❢❡rr❡❞ ✐♥t♦t❤❡ ♣❡r♣❡♥❞✐❝✉❧❛r ❜✐s❡❝t♦r ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s✐❞❡✳ ❚❤✐s ♠❡❛♥s t❤❛t ▼✬❂❖♣r♦✈✐♥❣ t❤❡ st❛t❡♠❡♥t✳

✹✳✷✳ ❚❍❊ ❊❯▲❊❘ ▲■◆❊ ❆◆❉ ❚❍❊ ❋❊❯❊❘❇❆❈❍ ❈■❘❈▲❊ ✼✸

❋✐❣✉r❡ ✹✳✽✿ ❋❡✉❡r❜❛❝❤✲❝✐r❝❧❡

❉❡✜♥✐t✐♦♥ ❚❤❡ ✐♠❛❣❡ ♦❢ t❤❡ ❝✐r❝✉♠s❝r✐❜❡❞ ❝✐r❝❧❡ ✉♥❞❡r t❤❡ s✐♠✐❧❛r✐t② ✇✐t❤❝❡♥t❡r ❙ ❛♥❞ r❛t✐♦ ✲ ✶✴✷ ✐s ❝❛❧❧❡❞ t❤❡ ❋❡✉❡r❜❛❝❤ ❝✐r❝❧❡ ♦❢ t❤❡ tr✐❛♥❣❧❡✳

❚❤❡♦r❡♠ ✹✳✷✳✷ ❚❤❡ ❋❡✉❡r❜❛❝❤ ❝✐r❝❧❡ ♣❛ss❡s t❤r♦✉❣❤ ♥✐♥❡ ♣♦✐♥ts✿

• t❤❡ ♠✐❞♣♦✐♥ts ♦❢ t❤❡ s✐❞❡s✱

• t❤❡ ❧❡❣s ♦❢ t❤❡ ❛❧t✐t✉❞❡s✱

• t❤❡ ♠✐❞♣♦✐♥ts ♦❢ t❤❡ s❡❣♠❡♥ts ❥♦✐♥✐♥❣ t❤❡ ♦rt❤♦❝❡♥t❡r ❛♥❞ t❤❡ ✈❡rt✐❝❡s❆✱ ❇ ❛♥❞ ❈✳

Pr♦♦❢ ❚❤❡ ❋❡✉❡r❜❛❝❤ ❝✐r❝❧❡ ♣❛ss❡s t❤r♦✉❣❤ t❤❡ ♠✐❞♣♦✐♥ts ♦❢ t❤❡ s✐❞❡s ❜❡✲❝❛✉s❡ t❤❡ ❝✐r❝✉♠s❝r✐❜❡❞ ❝✐r❝❧❡ ♣❛ss❡s t❤r♦✉❣❤ t❤❡ ✈❡rt✐❝❡s✳ ❚❤❡ r❛❞✐✉s ❘✬♦❢ t❤❡ ❋❡✉❡r❜❛❝❤ ❝✐r❝❧❡ ✐s ❥✉st ❘✴✷ ❜❡❝❛✉s❡ ♦❢ t❤❡ s✐♠✐❧❛r✐t② r❛t✐♦✳ ❙✐♥❝❡ ❙❞✐✈✐❞❡s t❤❡ s❡❣♠❡♥t ▼❖ ✐♥ t❤❡ r❛t✐♦ ✷ ✿ ✶ t❤❡ ❝❡♥t❡r ❖✬ ♦❢ t❤❡ ❋❡✉❡r❜❛❝❤❝✐r❝❧❡ ✐s t❤❡ ♠✐❞♣♦✐♥t ♦❢ t❤❡ s❡❣♠❡♥t ▼❖✳ ❚❤❡r❡❢♦r❡ ❖✬● ✐s t❤❡ ♠✐❞❧✐♥❡ ♦❢t❤❡ tr❛♣❡③♦✐❞ ❉▼❖❊ ❛♥❞ ● ❜✐s❡❝ts t❤❡ s❡❣♠❡♥t ❉❊✳ ❚❤✐s ♠❡❛♥s t❤❛t ❉❖✬❊✐s ❛♥ ✐s♦s❝❡❧❡s tr✐❛♥❣❧❡ ✇✐t❤

O′D = O′E = R/2

❛♥❞ t❤❡ ❧❡❣ ♣♦✐♥t ❉ ♦❢ t❤❡ ❛❧t✐t✉❞❡ ❜❡❧♦♥❣✐♥❣ t♦ t❤❡ s✐❞❡ ❝ ✐s ♦♥ t❤❡ ❋❡✉❡r❜❛❝❤❝✐r❝❧❡✳ ❋✐♥❛❧❧② ❖✬❋ ✐s ❛ ♠✐❞❧✐♥❡ ✐♥ t❤❡ tr✐❛♥❣❧❡ ❈▼❖✳ ❚❤❡r❡❢♦r❡

O′F =1

2CO = R/2

❛s ✇❛s t♦ ❜❡ st❛t❡❞✳

✼✹ ❈❍❆P❚❊❘ ✹✳ ❚❘■❆◆●▲❊❙

❋✐❣✉r❡ ✹✳✾✿ ❚r✐❛♥❣❧❡s

✹✳✸ ❙♣❡❝✐❛❧ tr✐❛♥❣❧❡s

❚r✐❛♥❣❧❡s ❝❛♥ ❜❡ ❝❧❛ss✐✜❡❞ ❜② ❛♥❣❧❡s ♦r s✐❞❡s✳ ■♥ ✇❤❛t ❢♦❧❧♦✇s ✇❡ s❤❛❧❧ ✉s❡t❤❡ ❜❛s✐❝ ♥♦t❛t✐♦♥s

• A✱ B ❛♥❞ C ❢♦r t❤❡ ✈❡rt✐❝❡s✱

• α✱ β ❛♥❞ γ ❢♦r t❤❡ ❛♥❣❧❡s ❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✈❡rt✐❝❡s ❛♥❞

• a✱ b ❛♥❞ c ❢♦r t❤❡ ♦♣♣♦s✐t❡ s✐❞❡s t♦ t❤❡ ❛♥❣❧❡s α✱ β ❛♥❞ γ✱ r❡s♣❡❝t✐✈❡❧②✳

❚❤❡ ♠♦st ✐♠♣♦rt❛♥t ❝❛s❡s ♦❢ s♣❡❝✐❛❧ tr✐❛♥❣❧❡s ❛r❡

• ❡q✉✐❧❛t❡r❛❧ ✭r❡❣✉❧❛r✮ tr✐❛♥❣❧❡s✿ ❛❧❧ s✐❞❡s ❛♥❞ ❛❧❧ ❛♥❣❧❡s ❛r❡ ❡q✉❛❧ t♦ ❡❛❝❤♦t❤❡r✱

• ✐s♦s❝❡❧❡s tr✐❛♥❣❧❡s✿ t✇♦ s✐❞❡s ❛♥❞ t❤❡ ♦♣♣♦s✐t❡ ❛♥❣❧❡s ❛r❡ ❡q✉❛❧ t♦ ❡❛❝❤♦t❤❡r✱

• ❛❝✉t❡ tr✐❛♥❣❧❡s✿ ❛❧❧ ❛♥❣❧❡s ❛r❡ ❧❡ss t❤❛♥ ✾✵ ❞❡❣r❡❡✱

• r✐❣❤t tr✐❛♥❣❧❡s✿ ♦♥❡ ♦❢ t❤❡ ❛♥❣❧❡ ✐s ✾✵ ❞❡❣r❡❡ ✐♥ ♠❡❛s✉r❡✱

• ♦❜t✉s❡ tr✐❛♥❣❧❡✿ ♦♥❡ ♦❢ t❤❡ ❛♥❣❧❡ ✐s ❣r❡❛t❡r t❤❛♥ ✾✵ ❞❡❣r❡❡

♦r ♠✐①❡❞ ❝❛s❡s✿ ❢♦r ❡①❛♠♣❧❡ ✐s♦s❝❡❧❡s r✐❣❤t tr✐❛♥❣❧❡s✳ ❖♥❡ ♦❢ t❤❡ ♦❧❞❡st ❢❛❝t✐♥ ❣❡♦♠❡tr② ✐s P②t❤❛❣♦r❡❛♥ t❤❡♦r❡♠ ❢♦r r✐❣❤t tr✐❛♥❣❧❡s✳

❚❤❡♦r❡♠ ✹✳✸✳✶ ✭P②t❤❛❣♦r❛s✱ ✺✼✵ ❇❈ ✲ ✹✾✺ ❇❈✮ ❚❤❡ s✉♠ ♦❢ t❤❡ sq✉❛r❡s ♦❢t❤❡ ❧❡❣s ✐s ❥✉st t❤❡ sq✉❛r❡ ♦❢ t❤❡ ❤②♣♦t❤❡♥✉s❡✿

a2 + b2 = c2.

✹✳✸✳ ❙P❊❈■❆▲ ❚❘■❆◆●▲❊❙ ✼✺

❋✐❣✉r❡ ✹✳✶✵✿ P②t❤❛❣♦r❡❛♥ t❤❡♦r❡♠✳

Pr♦♦❢ ■❢ ✇❡ ❞✐✈✐❞❡ ❛ sq✉❛r❡ ✇✐t❤ s✐❞❡s ♦❢ ❧❡♥❣t❤ ❛✰❜ ✐♥t♦ ✜✈❡ ♣❛rts ❜② t❤❡✜❣✉r❡ t❤❡♥ t❤❡ ❛r❡❛ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❛s

(a+ b)2 = 4ab

2+ c2.

P②t❤❛❣♦r❡❛♥ t❤❡♦r❡♠ ❢♦❧❧♦✇s ✐♠♠❡❞✐❛t❡❧② ❜② t❤❡ ❤❡❧♣ ♦❢ ❛♥ ❛❧❣❡❜r❛✐❝ ❝❛❧❝✉✲❧❛t✐♦♥✳

❘❡♠❛r❦ ❚❤❡ ♠❡❛♥✐♥❣ ♦❢ ❤②♣♦t❤❡♥✉s❡ ✐s str❡t❝❤❡❞✳ ❚❤❡ ✇♦r❞ r❡❢❡rs t♦ t❤❡❛♥❝✐❡♥t ♠❡t❤♦❞ t♦ ❝r❡❛t❡ r✐❣❤t ❛♥❣❧❡s ❜② ❛ s❡❣♠❡♥t❛❧ str✐♥❣ ✐♥ r❛t✐♦ ✸ ✿ ✹ ✿✺✳ ◆♦t❡ t❤❛t

32 + 42 = 52.

❚❤❡♦r❡♠ ✹✳✸✳✷ ✭❍❡✐❣❤t t❤❡♦r❡♠✮ ■❢ ♠ ❞❡♥♦t❡s t❤❡ ❛❧t✐t✉❞❡ ❜❡❧♦♥❣✐♥❣ t♦ t❤❡❤②♣♦t❤❡♥✉s❡ ✐♥ ❛ r✐❣❤t tr✐❛♥❣❧❡ t❤❡♥ m2 = pq✱ ✇❤❡r❡ ♣ ❛♥❞ q ❛r❡ t❤❡ ❧❡♥❣t❤s♦❢ t❤❡ s❡❣♠❡♥ts ❢r♦♠ t❤❡ ✈❡rt✐❝❡s t♦ t❤❡ ❧❡❣ ♣♦✐♥t ♦❢ t❤❡ ❛❧t✐t✉❞❡✳

Pr♦♦❢ ❇② P②t❤❛❣♦r❡❛♥ t❤❡♦r❡♠ ✐♥ t❤❡ tr✐❛♥❣❧❡s ❈❚❇✱ ❈❚❆ ❛♥❞ ❆❇❈

p2 +m2 = a2, q2 +m2 = b2, a2 + b2 = c2.

❚❤❡r❡❢♦r❡

p2 + q2 + 2m2 = a2 + b2 = c2 = (p+ q)2 = p2 + q2 + 2pq

✇❤✐❝❤ ♠❡❛♥s t❤❛t m2 = pq✳

✼✻ ❈❍❆P❚❊❘ ✹✳ ❚❘■❆◆●▲❊❙

❋✐❣✉r❡ ✹✳✶✶✿ ❍❡✐❣❤t t❤❡♦r❡♠✳

❘❡♠❛r❦ ■♥ ♦t❤❡r ✇♦r❞s t❤❡ ❛❧t✐t✉❞❡ ♠ ✐s t❤❡ ❣❡♦♠❡tr✐❝ ♠❡❛♥ ♦❢ ♣ ❛♥❞ q✳

❚❤❡♦r❡♠ ✹✳✸✳✸ ✭▲❡❣ t❤❡♦r❡♠s✮ a2 = cp ❛♥❞ b2 = cq✳

Pr♦♦❢ ❆s ❛❜♦✈❡p2 +m2 = a2 ❛♥❞ q2 +m2 = b2,

✇❤❡r❡ ♠ ✐s t❤❡ ❣❡♦♠❡tr✐❝ ♠❡❛♥ ♦❢ ♣ ❛♥❞ q✳ ❚❤❡r❡❢♦r❡ ✭❢♦r ❡①❛♠♣❧❡✮

a2 = p2 +m2 = p2 + pq = p(p+ q) = pc

❛s ✇❛s t♦ ❜❡ st❛t❡❞✳

❘❡♠❛r❦ ❚❤✐s ❝♦❧❧❡❝t✐♦♥ ♦❢ t❤❡♦r❡♠s ✭P②t❤❛❣♦r❡❛♥✱ ❍❡✐❣❤t ❛♥❞ ▲❡❣ t❤❡✲♦r❡♠s✮ ❛r❡ ♦❢t❡♥ r❡❢❡rr❡❞ ❛s s✐♠✐❧❛r✐t② t❤❡♦r❡♠s ✐♥ r✐❣❤t tr✐❛♥❣❧❡s ❜❡❝❛✉s❡t❤❡r❡ ❛r❡ ❛❧t❡r♥❛t✐✈❡ ♣r♦♦❢s ❜② ✉s✐♥❣ t❤❡ s✐♠✐❧❛r tr✐❛♥❣❧❡s ❈❚❇✱ ❈❚❆ ❛♥❞❆❇❈✳

❚❤❡♦r❡♠ ✹✳✸✳✹ ✭❚❤❛❧❡s t❤❡♦r❡♠✮ ■❢ ❆✱ ❇ ❛♥❞ ❈ ❛r❡ t❤r❡❡ ❞✐✛❡r❡♥t ♣♦✐♥ts♦♥ t❤❡ ♣❡r✐♠❡t❡r ♦❢ ❛ ❝✐r❝❧❡ s✉❝❤ t❤❛t ❆❇ ✐s ♦♥❡ ♦❢ t❤❡ ❞✐❛❣♦♥❛❧s t❤❡♥ ❆❇❈✐s ❛ r✐❣❤t tr✐❛♥❣❧❡ ❤❛✈✐♥❣ t❤❡ ❛♥❣❧❡ ♦❢ ♠❡❛s✉r❡ ✾✵ ❞❡❣r❡❡ ❛t ❈✳

Pr♦♦❢ ▲❡t ❖ ❜❡ t❤❡ ❝❡♥t❡r ♦❢ t❤❡ ❝✐r❝❧❡✳ ❙✐♥❝❡

OA = OB = OC = r

✐t ❢♦❧❧♦✇s t❤❛t ❆❖❈ ❛♥❞ ❇❖❈ ❛r❡ ✐s♦s❝❡❧❡s tr✐❛♥❣❧❡s✳ ❚❤❡r❡❢♦r❡

6 OAC = 6 OCA = α, 6 OBC = 6 OCB = β

❛♥❞✱ ❝♦♥s❡q✉❡♥t❧②✱

2(α + β) = 180 ⇒ α + β = 90.

✹✳✹✳ ❊❳❊❘❈■❙❊❙ ✼✼

❋✐❣✉r❡ ✹✳✶✷✿ ❚❤❛❧❡s t❤❡♦r❡♠✳

❘❡♠❛r❦ ❚❤❛❧❡s t❤❡♦r❡♠ ✐s ❛❝t✉❛❧❧② t❤❡ s♣❡❝✐❛❧ ❝❛s❡ ♦❢ ❛ ♠♦r❡ ❣❡♥❡r❛❧ ♦❜✲s❡r✈❛t✐♦♥ ❝❛❧❧❡❞ ✐♥s❝r✐❜❡❞ ❛♥❣❧❡ t❤❡♦r❡♠✿ ❧❡t ❆✱ ❇ ❛♥❞ ❈ ❜❡ t❤r❡❡ ❞✐✛❡r❡♥t♣♦✐♥ts ♦♥ t❤❡ ♣❡r✐♠❡t❡r ♦❢ ❛ ❝✐r❝❧❡ ✇✐t❤ ❝❡♥t❡r ❖ ❛♥❞ s✉♣♣♦s❡ t❤❛t t❤❡ ❛♥❣❧❡s6 AOB ❛♥❞ 6 ACB ❛r❡ ❧②✐♥❣ ♦♥ t❤❡ s❛♠❡ ❛r❝✳ ❚❤❡♥

6 AOB = 2 6 ACB

❜❡❝❛✉s❡6 AOB = ω = 2α + 2β = 2(α + β) = 2 6 ACB.

❚❤❡ ♣r♦♦❢ ✐s ❜❛s❡❞ ♦♥ t❤❡ ✐s♦s❝❡❧❡s tr✐❛♥❣❧❡s ❆❖❈ ❛♥❞ ❇❖❈✳

✹✳✹ ❊①❡r❝✐s❡s

❊①❝❡r❝✐s❡ ✹✳✹✳✶ ❈♦❧❧❡❝t t❤❡ ❢❛❝ts ✇❡ ✉s❡❞ t♦ ♣r♦✈❡ P②t❤❛❣♦r❡❛♥ t❤❡♦r❡♠✳

❙♦❧✉t✐♦♥✳ ❚❤❡ ♣r♦♦❢ ♦❢ P②t❤❛❣♦r❡❛♥ t❤❡♦r❡♠ ✐s ❜❛s❡❞ ♦♥

• t❤❡ ❛r❡❛ ♦❢ sq✉❛r❡s✱ r✐❣❤t tr✐❛♥❣❧❡s ❛♥❞ t❤❡ ❜❛s✐❝ ♣r✐♥❝✐♣❧❡s ♦❢ ♠❡❛s✉r✐♥❣t❤❡ ❛r❡❛✱

• t❤❡ s✉♠ ♦❢ ❛♥❣❧❡s ✐♥ ❛ ✭r✐❣❤t✮ tr✐❛♥❣❧❡ ✐s ✶✽✵ ❞❡❣r❡❡✱

• ❛❧❣❡❜r❛✐❝ ✐❞❡♥t✐t✐❡s✳

❊①❝❡r❝✐s❡ ✹✳✹✳✷ Pr♦✈❡ t❤❡ ❤❡✐❣❤t ❛♥❞ t❤❡ ❧❡❣ t❤❡♦r❡♠s ❜② ✉s✐♥❣ s✐♠✐❧❛r tr✐✲❛♥❣❧❡s✳ ❈♦♥❝❧✉❞❡ P②t❤❛❣♦r❡❛♥ t❤❡♦r❡♠ t♦♦✳

✼✽ ❈❍❆P❚❊❘ ✹✳ ❚❘■❆◆●▲❊❙

❋✐❣✉r❡ ✹✳✶✸✿ ■♥s❝r✐❜❡❞ ❛♥❣❧❡ t❤❡♦r❡♠

❊①❝❡r❝✐s❡ ✹✳✹✳✸ ❋✐♥❞ t❤❡ ♠✐ss✐♥❣ q✉❛♥t✐t✐❡s ✐♥ ❡❛❝❤ r♦✇ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣t❛❜❧❡✳

❛ ❜ ❝ ♠ ♣ q✶✷ ✸

✹ ✶✻✻ ✾

✻ ✾✻ ✽

❍✐♥t✳ ❯s❡ P②t❤❛❣♦r❡❛♥✱ ❤❡✐❣❤t ❛♥❞ ❧❡❣ t❤❡♦r❡♠s✿

a2 + b2 = c2, m2 = pq, a2 = cp ❛♥❞ b2 = cq.

❊①❝❡r❝✐s❡ ✹✳✹✳✹ ❋✐♥❞ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ s✐❞❡ ♦❢ ❛ r❡❣✉❧❛r tr✐❛♥❣❧❡ ✐♥s❝r✐❜❡❞✐♥ t❤❡ ✉♥✐t ❝✐r❝❧❡✳

❍✐♥t✳ ❯s✐♥❣ ❚❤❛❧❡s t❤❡♦r❡♠ t❤❡ tr✐❛♥❣❧❡ ABC ✐♥ t❤❡ ✜❣✉r❡ ❤❛s ❛ r✐❣❤t❛♥❣❧❡ ❛t t❤❡ ✈❡rt❡① C✳ ❚❤❡r❡❢♦r❡

x2 + 12 = 22,

✐✳❡✳ x =√3✳

❊①❝❡r❝✐s❡ ✹✳✹✳✺ ■♥ ❛ r✐❣❤t tr✐❛♥❣❧❡ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❧♦♥❣❡st s✐❞❡ AB ✐s 6✳❚❤❡ ❧❡❣ BC ✐s 3✳

✹✳✺✳ ❚❘■●❖◆❖▼❊❚❘❨ ✼✾

❋✐❣✉r❡ ✹✳✶✹✿ ❊①❡r❝✐s❡ ✹✳✹✳✹

• ❈❛❧❝✉❧❛t❡ t❤❡ ♠✐ss✐♥❣ ❧❡❣ ❛♥❞ t❤❡ ❛r❡❛ ♦❢ t❤❡ tr✐❛♥❣❧❡✳

• ❲❤❛t ✐s t❤❡ r❛❞✐✉s ♦❢ t❤❡ ✐♥s❝r✐❜❡❞ ❝✐r❝❧❡❄

• ❲❤❛t ❛r❡ t❤❡ s✐♥❡✱ ❝♦s✐♥❡✱ t❛♥❣❡♥t ❛♥❞ ❝♦t❛♥❣❡♥t ♦❢ t❤❡ ❛♥❣❧❡ ❛t ❆❄

✹✳✺ ❚r✐❣♦♥♦♠❡tr②

❊✉❝❧✐❞❡❛♥ ❣❡♦♠❡tr② ✐s ❡ss❡♥t✐❛❧❧② ❜❛s❡❞ ♦♥ tr✐❛♥❣❧❡s✳ ❚❤❡ ♠❡tr✐❝ ♣r♦♣❡rt✐❡s♦❢ tr✐❛♥❣❧❡s ✭t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ s✐❞❡s ♦r t❤❡ ♠❡❛s✉r❡ ♦❢ t❤❡ ❛♥❣❧❡s✮ ❝❛♥ ❜❡❞❡s❝r✐❜❡❞ ❜② ❡❧❡❣❛♥t ❢♦r♠✉❧❛s✳ ❚❤❡② ❛r❡ ✈❡r② ✐♠♣♦rt❛♥t ✐♥ ♣r❛❝t✐❝❡ t♦♦ ✭s❡❡❝❤❛♣t❡r ✻✮✳ ❚❤❡ ✇♦r❞ tr✐❣♦♥♦♠❡tr② ❞✐r❡❝t❧② ♠❡❛♥s t❤❡ ♠❡❛s✉r✐♥❣ ♦❢ tr✐❛♥❣❧❡s✳❯s✐♥❣ t❤❡ ❜❛s✐❝ ❝❛s❡s ♦❢ s✐♠✐❧❛r✐t② ✐t ❝❛♥ ❜❡ ❡❛s✐❧② s❡❡♥ t❤❛t t✇♦ r✐❣❤t tr✐❛♥✲❣❧❡s ✇✐t❤ ❛❝✉t❡ ❛♥❣❧❡s ♦❢ t❤❡ s❛♠❡ ♠❡❛s✉r❡ ❛r❡ s✐♠✐❧❛r✳ ❚❤❡r❡❢♦r❡ t❤❡ r❛t✐♦s❜❡t✇❡❡♥ t❤❡ ❧❡❣s ❛♥❞ t❤❡ ❤②♣♦t❤❡♥✉s ❛r❡ ✉♥✐q✉❡❧② ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ❛♥❣❧❡s✳❚❤✐s r❡s✉❧ts ✐♥ t❤❡ ♥♦t✐♦♥ ♦❢ s✐♥❡✱ ❝♦s✐♥❡✱ t❛♥❣❡♥t ❛♥❞ ❝♦t❛♥❣❡♥t ✐♥ t❤❡ ❢♦❧✲❧♦✇✐♥❣ ✇❛②✳ ▲❡t α ❜❡ ❛♥ ❛❝✉t❡ ❛♥❣❧❡✱ ✐✳❡✳ 0 < α < 90◦✳ ■❢ ❆❇❈ ✐s ❛ r✐❣❤ttr✐❛♥❣❧❡ ✇✐t❤ ❧❡❣s ❆❈ ❛♥❞ ❇❈ ❛♥❞ t❤❡ ❛♥❣❧❡ ❛t t❤❡ ❝♦r♥❡r ❆ ✐s α t❤❡♥

• t❤❡ s✐♥❡ ♦❢ α ✐s t❤❡ r❛t✐♦ ❜❡t✇❡❡♥ t❤❡ ♦♣♣♦s✐t❡ ❧❡❣ ❛♥❞ t❤❡ ❤②♣♦t❤❡♥✉s❡✿sinα = a/c,

• t❤❡ ❝♦s✐♥❡ ♦❢ α ✐s t❤❡ r❛t✐♦ ❜❡t✇❡❡♥ t❤❡ ❛❞❥❛❝❡♥t ❧❡❣ ❛♥❞ t❤❡ ❤②✲♣♦t❤❡♥✉s❡✿ cosα = b/c

✽✵ ❈❍❆P❚❊❘ ✹✳ ❚❘■❆◆●▲❊❙

❋✐❣✉r❡ ✹✳✶✺✿ ❚r✐❣♦♥♦♠❡tr② ✐♥ ❛ r✐❣❤t tr✐❛♥❣❧❡

• t❤❡ t❛♥❣❡♥t ♦❢ α ✐s t❤❡ r❛t✐♦ ❜❡t✇❡❡♥ t❤❡ ♦♣♣♦s✐t❡ ❛♥❞ t❤❡ ❛❞❥❛❝❡♥t ❧❡❣✿tanα = a/b✱

• t❤❡ ❝♦t❛♥❣❡♥t ♦❢ α ✐s t❤❡ r❛t✐♦ ❜❡t✇❡❡♥ t❤❡ ❛❞❥❛❝❡♥t ❧❡❣ ❛♥❞ t❤❡ ♦♣♣♦s✐t❡❧❡❣✿ cotα = b/a✳

❲❡ ❝❛♥ ❡❛s✐❧② ❝♦♥❝❧✉❞❡ t❤❛t

sinα = cos(90− α) ❛♥❞ cosα = sin(90− α),

sin2 α + cos2 α = 1

✭tr✐❣♦♥♦♠❡tr✐❝ P②t❤❛❣♦r❡❛♥ t❤❡♦r❡♠✮✱

tanα =sinα

cosα, cotα =

cosα

sinα, tanα =

1

cotα,

tanα = cot(90− α) ❛♥❞ cotα = tan(90− α).

■t ✐s ❤❛r❞ t♦ ❝r❡❛t❡ ❛ ❣❡♦♠❡tr✐❝ ❝♦♥✜❣✉r❛t✐♦♥ t♦ ✜♥❞ t❤❡ s✐♥❡ ❛♥❞ ❝♦s✐♥❡✭t❛♥❣❡♥t ❛♥❞ ❝♦t❛♥❣❡♥t✮ ♦❢ ❛ ❣✐✈❡♥ ❛♥❣❧❡ ✐♥ ❣❡♥❡r❛❧✳ ❚❤❡ s♦✲❝❛❧❧❡❞ ❛❞❞✐t✐♦♥❛❧r✉❧❡s ❤❡❧♣ ✉s t♦ s♦❧✈❡ s✉❝❤ ❦✐♥❞ ♦❢ ♣r♦❜❧❡♠s✳

❚❤❡♦r❡♠ ✹✳✺✳✶ ✭❆❞❞✐t✐♦♥❛❧ r✉❧❡s✮

sin(α + β) = sinα cos β + cosα sin β,

cos(α + β) = cosα cos β − sinα sin β.

❙♣❡❝✐❛❧ ❝❛s❡ ❛r❡

sin 2α = 2 sinα cosα,

cos 2α = cos2 α− sin2 α.

✹✳✺✳ ❚❘■●❖◆❖▼❊❚❘❨ ✽✶

❋✐❣✉r❡ ✹✳✶✻✿ ❆❞❞✐t✐♦♥❛❧ r✉❧❡s

Pr♦♦❢ ❯s✐♥❣ t❤❡ ♥♦t❛t✐♦♥s ✐♥ t❤❡ ✜❣✉r❡ ✇❡ ✜♥❞ t❤❛t

sin(α + β) =DF

DO.

❋♦r t❤❡ s❛❦❡ ♦❢ s✐♠♣❧✐❝✐t② s✉♣♣♦s❡ t❤❛t ❉❖ ❂ ❇❖ ❂ ✶✳ ❚❤❡r❡❢♦r❡

sin(α + β) = DF = DE + EF = DE + CG = CD cosα + CO sinα =

sin β cosα + cos β sinα.

❖♥ t❤❡ ♦t❤❡r ❤❛♥❞

cos(α + β) = OF = OG−GF = OG− CE = CO cosα− CD sinα =

cos β cosα− sin β sinα

❛s ✇❛s t♦ ❜❡ ♣r♦✈❡❞✳

❚❤❡ ❛❞❞✐t✐♦♥❛❧ r✉❧❡s ❝❛♥ ❜❡ ✉s❡❞ t♦ ❡①t❡♥❞ t❤❡ ♥♦t✐♦♥ ♦❢ s✐♥❡✱ ❝♦s✐♥❡✱ t❛♥❣❡♥t❛♥❞ ❝♦t❛♥❣❡♥t✳ ❯s✐♥❣ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ✾✵❂✹✺✰✹✺ ✇❡ ❤❛✈❡ ✐♠♠❡❞✐❛t❡❧②t❤❛t

sin 90 = 2 sin 45 cos 45 = 1 ❛♥❞ cos 90 = cos2 45− sin2 45 = 0.

❚❤❡ ❡①t❡♥s✐♦♥ ✐♥ ♠❛t❤❡♠❛t✐❝s ✐s ✉s✉❛❧❧② ❜❛s❡❞ ♦♥ t❤❡ ♣r✐♥❝✐♣❧❡ ♦❢ ♣❡r♠❛✲♥❡♥❝❡✳ ❚❤✐s ♠❡❛♥s t❤❛t ✇❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ❦❡❡♣ ❛❧❧ t❤❡ ♣r❡✈✐♦✉s r✉❧❡s ✭❝❢✳ t❤❡❡①t❡♥s✐♦♥ ♦❢ ♣♦✇❡rs ❢r♦♠ ♥❛t✉r❛❧s t♦ r❛t✐♦♥❛❧s✮✳ ❆s ❛♥♦t❤❡r ❡①❛♠♣❧❡ ❝♦♠♣✉t❡sin 105 ✇✐t❤ t❤❡ ❤❡❧♣ ♦❢ ❞❡❝♦♠♣♦s✐t✐♦♥ ✶✵✺❂✻✵✰✹✺✿

sin 105 = sin(60 + 45) = sin 60 cos 45 + cos 60 sin 45 =

√3

2

1√2+

1

2

1√2.

✽✷ ❈❍❆P❚❊❘ ✹✳ ❚❘■❆◆●▲❊❙

◆❡✇ r❡❧❛t✐♦♥s❤✐♣s ❝❛♥ ❜❡ ❝r❡❛t❡❞ s✉❝❤ ❛s

sin(α + 90) = sinα cos 90 + cosα sin 90 = cosα,

cos(α + 90) = cosα cos 90− sinα sin 90 = − sinα.

❊s♣❡❝✐❛❧❧② t❤❡ s✐♥❡ ✐s ♣♦s✐t✐✈❡ ✐♥ t❤❡ s❡❝♦♥❞ q✉❛❞r❛♥t ♦❢ t❤❡ ♣❧❛♥❡ ❜✉t t❤❡❝♦s✐♥❡ ❤❛s ❛ ♠✐♥✉s s✐❣♥✳ ■♥ ❛ s✐♠✐❧❛r ✇❛②

sin 180 = 2 sin 90 cos 90 = 0 ❛♥❞ cos 180 = cos2 90− sin2 90 = −1.

❚❤❡r❡❢♦r❡

sin(α + 180) = − sinα ❛♥❞ cos(α + 180) = − cosα,

✐✳❡✳ ❜♦t❤ t❤❡ s✐♥❡ ❛♥❞ t❤❡ ❝♦s✐♥❡ ❛r❡ ♥❡❣❛t✐✈❡ ✐♥ t❤❡ t❤✐r❞ q✉❛❞r❛♥t✳ ❚♦✐♥✈❡st✐❣❛t❡ t❤❡ ❢♦✉rt❤ q✉❛❞r❛♥t ✈❡r✐❢② t❤❛t

cos 270 = 0 ❛♥❞ sin 270 = −1.

❲❡ ❤❛✈❡ t❤❛t t❤❡ s✐♥❡ ✐s ♥❡❣❛t✐✈❡ ✐♥ t❤❡ ❧❛st q✉❛❞r❛♥t ❜✉t t❤❡ ❝♦s✐♥❡ ❦❡❡♣s✐ts ♣♦s✐t✐✈❡ s✐❣♥ ❜❡❝❛✉s❡ ♦❢

sin(α + 270) = − cosα < 0 ❛♥❞ cos(α + 270) = sinα > 0

❢♦r ❛♥② ❛❝✉t❡ ❛♥❣❧❡ α✳ ❋✐♥❛❧❧②

sin(α + 360) = sinα ❛♥❞ cos(α + 360) = cosα.

❚❤❡ ♣❡r✐♦❞✐❝✐t② ♣r♦♣❡rt✐❡s s❤♦✇ t❤❛t t❤❡ ♣r♦❝❡ss ♦❢ ❡①t❡♥s✐♦♥ ❣♦❡s t♦ t❤❡❡♥❞✳ ❋r♦♠ ♥♦✇ ♦♥ tr✐❣♦♥♦♠❡tr✐❝ ❡①♣r❡ss✐♦♥s ❝❛♥ ❜❡ ❝♦♥s✐❞❡r❡❞ ❛s ❢✉♥❝t✐♦♥s❬✻❪✳ ❚❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ s✐♥❡ ❛♥❞ ❝♦s✐♥❡ ❢✉♥❝t✐♦♥s ❛r❡ t❤❡ s❡t ♦❢ ❛❧❧ ❛♥❣❧❡s♠❡❛s✉r❡❞ ✐♥ ❞❡❣r❡❡ ♦r r❛❞✐❛♥✳ ■♥ ♠❛t❤❡♠❛t✐❝s t❤❡ r❛❞✐❛♥ ✐s ♠♦r❡ t②♣✐❝❛❧❜❡❝❛✉s❡ ✐t ✐s ❞✐r❡❝t❧② r❡❧❛t❡❞ t♦ t❤❡ ❣❡♦♠❡tr✐❝ ❧❡♥❣t❤ ♦❢ t❤❡ ❛r❝ ❛❧♦♥❣ ❛ ✉♥✐t❝✐r❝❧❡ ✭❛ ❝✐r❝❧❡ ❤❛✈✐♥❣ r❛❞✐✉s ♦♥❡✮✳ ❚❤❡ ❛♥❣❧❡ ❜❡❧♦♥❣✐♥❣ t♦ t❤❡ ❛r❝ ♦❢ ✉♥✐t❧❡♥❣t❤ ✐s ✶ r❛❞✐❛♥ ✐♥ ♠❡❛s✉r❡✳ ❚❤❡ r❡❧❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ t❤❡ ❞❡❣r❡❡ ❛♥❞ t❤❡r❛❞✐❛♥ ✐s ❥✉st

α ✭❞❡❣✮360

=α ✭r❛❞✮

2π.

❘❡♠❛r❦ ❚♦ ♠❡♠♦r✐③❡ t❤❡ s✐❣♥s ♦❢ tr✐❣♦♥♦♠❡tr✐❝ ❡①♣r❡ss✐♦♥s ❝♦♥s✐❞❡r t❤❡♠♦t✐♦♥ ♦❢ ❛ ♣♦✐♥t ❛❧♦♥❣ t❤❡ ✉♥✐t ❝✐r❝❧❡ ❝❡♥t❡r❡❞ ❛t t❤❡ ♦r✐❣✐♥ ✐♥ t❤❡ ❊✉✲❝❧✐❞❡❛♥ ❝♦♦r❞✐♥❛t❡ ♣❧❛♥❡✳ ❚❤❡ ❝♦s✐♥❡ ❛♥❞ t❤❡ s✐♥❡ ❢✉♥❝t✐♦♥s ❣✐✈❡ t❤❡ ✜rst❛♥❞ t❤❡ s❡❝♦♥❞ ❝♦♦r❞✐♥❛t❡s ✐♥ t❡r♠s ♦❢ t❤❡ r♦t❛t✐♦♥❛❧ ❛♥❣❧❡✳ ❖❜✈✐♦✉s❧② ✇❡❤❛✈❡ ♣♦s✐t✐✈❡ ❝♦♦r❞✐♥❛t❡s ✐♥ t❤❡ ✜rst q✉❛❞r❛♥t✳ ❆❢t❡r ❡♥t❡r✐♥❣ ✐♥ t❤❡ s❡❝♦♥❞q✉❛❞r❛♥t t❤❡ ✜rst ❝♦♦r❞✐♥❛t❡ ♠✉st ❜❡ ♥❡❣❛t✐✈❡ ❛♥❞ s♦ ♦♥✳ ❋♦r t❤❡ ✐❧❧✉str❛t✐♦♥♦❢ t❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s s❡❡ ✜❣✉r❡s ✹✳✶✼ ❛♥❞ ✹✳✶✽✳

✹✳✺✳ ❚❘■●❖◆❖▼❊❚❘❨ ✽✸

❋✐❣✉r❡ ✹✳✶✼✿ ❚❤❡ s✐♥❡ ❢✉♥❝t✐♦♥

❋✐❣✉r❡ ✹✳✶✽✿ ❚❤❡ t❛♥❣❡♥t ❢✉♥❝t✐♦♥

✽✹ ❈❍❆P❚❊❘ ✹✳ ❚❘■❆◆●▲❊❙

❋✐❣✉r❡ ✹✳✶✾✿ ❊①❡r❝✐s❡ ✹✳✻✳✷

✹✳✻ ❊①❡r❝✐s❡s

❊①❝❡r❝✐s❡ ✹✳✻✳✶ ❈♦♠♣✉t❡ t❤❡ ❡①❛❝t ✈❛❧✉❡s ♦❢ s✐♥❡✱ ❝♦s✐♥❡✱ t❛♥❣❡♥t ❛♥❞ ❝♦t❛♥✲❣❡♥t ❢✉♥❝t✐♦♥s ❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ ❛♥❣❧❡s✿

45, 30, 60.

❙♦❧✉t✐♦♥✳ ❋r♦♠ ❛♥ ✐s♦s❝❡❧❡s r✐❣❤t tr✐❛♥❣❧❡ ✇❡ ❤❛✈❡ t❤❛t

sin 45 = cos 45 =1√2.

❋r♦♠ ❛ r❡❣✉❧❛r tr✐❛♥❣❧❡ ✇✐t❤ s✐❞❡s ♦❢ ✉♥✐t ❧❡♥❣t❤ ✇❡ ❤❛✈❡ t❤❛t

cos 60 = sin 30 =1

2

❛♥❞

sin 60 = cos 30 =

√3

2.

❊①❝❡r❝✐s❡ ✹✳✻✳✷ ❈♦♠♣✉t❡ t❤❡ ❡①❛❝t ✈❛❧✉❡s ♦❢ s✐♥❡✱ ❝♦s✐♥❡✱ t❛♥❣❡♥t ❛♥❞ ❝♦t❛♥✲❣❡♥t ❢✉♥❝t✐♦♥s ❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ ❛♥❣❧❡s✿

72, 36, 18.

❙♦❧✉t✐♦♥✳ ❈♦♥s✐❞❡r ❛ r❡❣✉❧❛r ✶✵✕❣♦♥ ✐♥s❝r✐❜❡❞ ✐♥ t❤❡ ✉♥✐t ❝✐r❝❧❡✳ ❆s t❤❡ ✜❣✉r❡s❤♦✇s t❤❡ tr✐❛♥❣❧❡s OAB ❛♥❞ DOA ❛r❡ s✐♠✐❧❛r ✇❤✐❝❤ ♠❡❛♥s t❤❛t

1 : x = (1 + x) : 1,

✹✳✻✳ ❊❳❊❘❈■❙❊❙ ✽✺

✇❤❡r❡ x ❞❡♥♦t❡s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ s✐❞❡ ❆❇✳ ❲❡ ❤❛✈❡ ❛ q✉❛❞r❛t✐❝ ❡q✉❛t✐♦♥

x2 + x− 1 = 0.

❚❤❡r❡❢♦r❡

x =−1 +

√5

2.

❚♦ ❡①♣r❡ss ✭❢♦r ❡①❛♠♣❧❡✮ cos 72 ❝♦♥s✐❞❡r t❤❡ ♣❡r♣❡♥❞✐❝✉❧❛r ❜✐s❡❝t♦r ♦❢ t❤❡s✐❞❡ ❆❇ ✐♥ t❤❡ tr✐❛♥❣❧❡ ❖❆❇✳ ❙✐♥❝❡ t❤❡ r❛❞✐✉s ✐s ✶ ✇❡ ❤❛✈❡ t❤❛t

cos 72 = x/2

❛♥❞✱ ❝♦♥s❡q✉❡♥t❧②✱sin 72 =

√

1− (x/2)2

❜② t❤❡ tr✐❣♦♥♦♠❡tr✐❝ P②t❤❛❣♦r❡❛♥ t❤❡♦r❡♠✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞

cos 18 = sin 72 ❛♥❞ sin 18 = cos 72.

❚♦ ❞❡t❡r♠✐♥❡ t❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❡①♣r❡ss✐♦♥s ♦❢ t❤❡ ❛♥❣❧❡ ✸✻ ❞❡❣r❡❡ ✐♥ ♠❡❛s✉r❡✉s❡ t❤❡ ♣❡r♣❡♥❞✐❝✉❧❛r ❜✐s❡❝t♦r ❜❡❧♦♥❣✐♥❣ t♦ t❤❡ s✐❞❡ ❖❉ ✐♥ t❤❡ tr✐❛♥❣❧❡ ❖❇❉✳❙✐♥❝❡ t❤❡ r❛❞✐✉s ♦❢ t❤❡ ❝✐r❝❧❡ ✐s ✶ ✇❡ ❤❛✈❡ t❤❛t

cos 36 = (1 + x)/2 ❛♥❞ sin 36 =√

1− (1 + x)2/4.

❊①❝❡r❝✐s❡ ✹✳✻✳✸ ❈♦♠♣✉t❡ t❤❡ ❡①❛❝t ✈❛❧✉❡s ♦❢ s✐♥❡✱ ❝♦s✐♥❡✱ t❛♥❣❡♥t ❛♥❞ ❝♦t❛♥✲❣❡♥t ❢✉♥❝t✐♦♥s ❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ ❛♥❣❧❡s✿

75, 54, 22.5.

❙♦❧✉t✐♦♥✳ ❯s✐♥❣ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥s

75 = 45 + 30, ❛♥❞ 54 = 36 + 18

t❤❡ ❛❞❞✐t✐♦♥❛❧ r✉❧❡s ❣✐✈❡ t❤❡ ✈❛❧✉❡s ♦❢ s✐♥❡✱ ❝♦s✐♥❡ t❛♥❣❡♥t ❛♥❞ ❝♦t❛♥❣❡♥t✳❋✐♥❛❧❧②

45 = 2 · 22.5❛♥❞

cos 45 = cos2 22.5− sin2 22.5 = 1− 2 sin2 22.5

❜❡❝❛✉s❡ ♦❢ t❤❡ tr✐❣♦♥♦♠❡tr✐❝ P②t❤❛❣♦r❡❛♥ t❤❡♦r❡♠

cos2 α + sin2 α = 1.

❚❤❡r❡❢♦r❡

sin 22.5 =

√

1− cos 45

2

❛♥❞ s♦ ♦♥✳

✽✻ ❈❍❆P❚❊❘ ✹✳ ❚❘■❆◆●▲❊❙

❋✐❣✉r❡ ✹✳✷✵✿ ❙✐♥❡ r✉❧❡ ✲ ❛❝✉t❡ ❛♥❣❧❡s

❊①❝❡r❝✐s❡ ✹✳✻✳✹ ❊①♣r❡ss

cos 3α, sin 3α, cos 4α, sin 4α, . . .

✐♥ t❡r♠s ♦❢ sinα ❛♥❞ cosα✳

❊①❝❡r❝✐s❡ ✹✳✻✳✺ ❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❝♦s✐♥❡ ❢✉♥❝t✐♦♥✳

❍✐♥t✳ ❯s❡ t❤❛tcosα = sin(α + 90).

❊①❝❡r❝✐s❡ ✹✳✻✳✻ ❊①♣❧❛✐♥ ✇❤❡r❡ t❤❡ ♥❛♠❡ t❛♥❣❡♥t ❝♦♠❡s ❢r♦♠❄

❊①❝❡r❝✐s❡ ✹✳✻✳✼ ❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❝♦t❛♥❣❡♥t ❢✉♥❝t✐♦♥✳

❍✐♥t✳ ❯s❡ t❤❛tcotα = tan(90− α) = − tan(α− 90).

✹✳✼ ●❡♥❡r❛❧ tr✐❛♥❣❧❡s ■■ ✲ ❙✐♥❡ ❛♥❞ ❈♦s✐♥❡ r✉❧❡

❖♥❡ ♦❢ t❤❡ ♠♦st ✐♠♣♦rt❛♥t ❛♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡ ❡①t❡♥❞❡❞ s✐♥❡ ❛♥❞ ❝♦s✐♥❡❢✉♥❝t✐♦♥s ✐s t♦ ❝♦♥❝❧✉❞❡ t❤❡ s✐♥❡ ❛♥❞ ❝♦s✐♥❡ r✉❧❡s ❢♦r ❣❡♥❡r❛❧ tr✐❛♥❣❧❡s✳

✹✳✼✳✶ ❙✐♥❡ r✉❧❡

❋✐rst ♦❢ ❛❧❧ ✇❡ ✐♥✈❡st✐❣❛t❡ t❤❡ ❝❛s❡ ♦❢ ❛❝✉t❡ tr✐❛♥❣❧❡s ✭❛❧❧ t❤❡ ❛♥❣❧❡s ❛r❡ ❧❡sst❤❛♥ ✾✵ ❞❡❣r❡❡ ✐♥ ♠❡❛s✉r❡✮✳ ❚♦ ♣r❡s❡♥t t❤❡ s✐♥❡ r✉❧❡ ❧❡t ✉s st❛rt ✇✐t❤ t❤❡❝✐r❝✉♠s❝r✐❜❡❞ ❝✐r❝❧❡ ♦❢ t❤❡ tr✐❛♥❣❧❡ ❆❇❈✳ ❚❤❡ ❝❡♥t❡r ✐s ❥✉st t❤❡ ✐♥t❡rs❡❝t✐♦♥♦❢ t❤❡ ♣❡r♣❡♥❞✐❝✉❧❛r ❜✐s❡❝t♦rs ♦❢ t❤❡ s✐❞❡s✳ ❙✐♥❝❡ ❇❖❈ ✐s ❛♥ ✐s♦s❝❡❧❡s tr✐❛♥❣❧❡t❤❡ ✐♥s❝r✐❜❡❞ ❛♥❣❧❡ t❤❡♦r❡♠ s❛②s t❤❛t

6 DOC = α,

✹✳✼✳ ●❊◆❊❘❆▲ ❚❘■❆◆●▲❊❙ ■■ ✲ ❙■◆❊ ❆◆❉ ❈❖❙■◆❊ ❘❯▲❊ ✽✼

❋✐❣✉r❡ ✹✳✷✶✿ ❙✐♥❡ r✉❧❡ ✲ ❛♥ ♦❜t✉s❡ ❛♥❣❧❡

✇❤❡r❡ ❉ ✐s t❤❡ ♠✐❞♣♦✐♥t ♦❢ ❇❈✳ ❚❤❡r❡❢♦r❡

sinα =a/2

R⇒ 2R =

a

sinα.

❚❤❡♦r❡♠ ✹✳✼✳✶ ✭❙✐♥❡ r✉❧❡✮

a

sinα=

b

sin β=

c

sin γ= 2R.

❊①❝❡r❝✐s❡ ✹✳✼✳✷ Pr♦✈❡ t❤❡ s✐♥❡ r✉❧❡ ✐♥ ❝❛s❡ ♦❢ ♦❜t✉s❡ tr✐❛♥❣❧❡s✳

❍✐♥t✳ ❖❜s❡r✈❡ t❤❛t sin(180− α) = sinα✳

✹✳✼✳✷ ❈♦s✐♥❡ r✉❧❡

❚❤❡ ❝♦s✐♥❡ r✉❧❡ ✐s t❤❡ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ P②t❤❛❣♦r❡❛♥ t❤❡♦r❡♠✳ ❆t ✜rst ✇❡❞✐s❝✉ss ❛❝✉t❡ tr✐❛♥❣❧❡s ❛❣❛✐♥✳ ❯s✐♥❣ t❤❡ ❛❧t✐t✉❞❡ ❜❡❧♦♥❣✐♥❣ t♦ t❤❡ s✐❞❡ ❜ ✇❡❡①♣r❡ss t❤❡ sq✉❛r❡ ♦❢ ❛ ✐♥ t✇♦ st❡♣s ❜② ✉s✐♥❣ P②t❤❛❣♦r❡❛♥ t❤❡♦r❡♠✳ ■❢ ❳ ✐st❤❡ ❢♦♦t ♣♦✐♥t ♦❢ t❤❡ ❛❧t✐t✉❞❡ t❤❡♥ ✇❡ ❝❛♥ ✇r✐t❡ t❤❛t

AX = c cosα ❛♥❞ BX = c sinα.

❚❤❡r❡❢♦r❡

a2 = BX2 +XC2 = BX2 + (CA− AX)2 = c2 sin2 α + (b− c cosα)2 =

c2 sin2 α + c2 cos2 α + b2 − 2bc cosα = c2 + b2 − 2bc cosα.

✽✽ ❈❍❆P❚❊❘ ✹✳ ❚❘■❆◆●▲❊❙

❋✐❣✉r❡ ✹✳✷✷✿ ❈♦s✐♥❡ r✉❧❡ ✲ ❛❝✉t❡ ❛♥❣❧❡s

❚❤❡♦r❡♠ ✹✳✼✳✸ ✭❈♦s✐♥❡ r✉❧❡✮

a2 = c2 + b2 − 2bc cosα.

b2 = c2 + a2 − 2ac cos β,

c2 = a2 + b2 − 2ab cos γ.

❊①❝❡r❝✐s❡ ✹✳✼✳✹ Pr♦✈❡ t❤❡ ❝♦s✐♥❡ r✉❧❡ ✐♥ ❝❛s❡ ♦❢ ♦❜t✉s❡ tr✐❛♥❣❧❡s✳

❍✐♥t✳ ❖❜s❡r✈❡ t❤❛t ✐❢ α > 90 t❤❡♥ t❤❡ ❢♦♦t ♣♦✐♥t ♦❢ t❤❡ ❛❧t✐t✉❞❡ ❜❡❧♦♥❣✐♥❣ t♦❜ ✐s ♦✉ts✐❞❡ ❢r♦♠ t❤❡ s❡❣♠❡♥t ❆❈✳ ❲❡ s❤♦✉❧❞ ✉s❡ t❤❡ ❛❝✉t❡ ❛♥❣❧❡ α′ = 180−αt♦ ❡①♣r❡ss ❆❳ ❛♥❞ ❇❳ ❛s ❛❜♦✈❡✿

AX = c cosα′ ❛♥❞ BX = c sinα′.

❚❤❡r❡❢♦r❡

a2 = BX2 +XC2 = BX2 + (CA+ AX)2 = c2 sin2 α′ + (b+ c cosα′)2 =

c2 + b2 + 2bc cosα′ = c2 + b2 − 2bc cosα

❜❡❝❛✉s❡ ♦❢cos(180− α) = − cosα.

✹✳✼✳✸ ❆r❡❛ ♦❢ tr✐❛♥❣❧❡s

■♥ ✇❤❛t ❢♦❧❧♦✇s ✇❡ s❤❛❧❧ ✉s❡ t❤❡ ❛①✐♦♠s ♦❢ ♠❡❛s✉r✐♥❣ ❛r❡❛❀ s❡❡ s❡❝t✐♦♥ ✸✳✺✳❚❤❡ ❛r❡❛ ♦❢ r✐❣❤t tr✐❛♥❣❧❡s✳ ❯s✐♥❣ t❤❡ ❛r❡❛ ❛❞❞✐t✐♦♥ ❛①✐♦♠ ✇❡ ❝❛♥ ❡❛s✐❧②❝♦♥❝❧✉❞❡ t❤❛t t❤❡ ❛r❡❛ ♦❢ ❛ r✐❣❤t tr✐❛♥❣❧❡ ✇✐t❤ ❧❡❣s ❛ ❛♥❞ ❜ ✐s ❥✉st ❛❜✴✷✳ ❚❤❡

✹✳✼✳ ●❊◆❊❘❆▲ ❚❘■❆◆●▲❊❙ ■■ ✲ ❙■◆❊ ❆◆❉ ❈❖❙■◆❊ ❘❯▲❊ ✽✾

❋✐❣✉r❡ ✹✳✷✸✿ ❈♦s✐♥❡ r✉❧❡ ✲ ❛♥ ♦❜t✉s❡ ❛♥❣❧❡

❛❧t✐t✉❞❡ ❜❡❧♦♥❣✐♥❣ t♦ t❤❡ ❤②♣♦t❤❡♥✉s❡ ❞✐✈✐❞❡s t❤❡ r✐❣❤t tr✐❛♥❣❧❡ ✐♥t♦ t✇♦ r✐❣❤ttr✐❛♥❣❧❡s✳ ❚❤❡r❡❢♦r❡ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛ t♦ ❝♦♠♣✉t❡ t❤❡ ❛r❡❛✿

A =pm

2+

qm

2=

(p+ q)m

2=

cm

2,

✇❤❡r❡ ♠ ❞❡♥♦t❡s t❤❡ ❛❧t✐t✉❞❡ ✭❤❡✐❣❤t✮ ❜❡❧♦♥❣✐♥❣ t♦ t❤❡ ❤②♣♦t❤❡♥✉s❡ ❝✳ ❚❤❡❧❡❣s ❛r❡ ✇♦r❦✐♥❣ ❛s ❛❧t✐t✉❞❡s ❜❡❧♦♥❣✐♥❣ t♦ ❡❛❝❤ ♦t❤❡r✳❚❤❡ ❛r❡❛ ♦❢ ❛ ❣❡♥❡r❛❧ tr✐❛♥❣❧❡ ❝❛♥ ❜❡ ❛❧s♦ ❝♦♠♣✉t❡❞ ❜② t❤❡ ❛r❡❛ ❛❞❞✐t✐♦♥❛①✐♦♠✳ ❚❤❡ ❜❛s✐❝ ❢♦r♠✉❧❛s t♦ ❝♦♠♣✉t❡ t❤❡ ❛r❡❛ ❛r❡

A =ama

2=

bmb

2=

cmc

2,

✇❤❡r❡ ma✱ mb ❛♥❞ mc ❞❡♥♦t❡ t❤❡ ❛❧t✐t✉❞❡s ❜❡❧♦♥❣✐♥❣ t♦ t❤❡ s✐❞❡s ❛✱ ❜ ❛♥❞❝✱ r❡s♣❡❝t✐✈❡❧②✳ ■♥ ♣r❛❝t✐❝❡ ✐t ✐s ✉s✉❛❧❧② ❤❛r❞ t♦ ♠❡❛s✉r❡ t❤❡ ❛❧t✐t✉❞❡ ✭✐✳❡✳t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ❛ ❧✐♥❡ ❛♥❞ ❛ ♣♦✐♥t✮ ✐♥ ❛ ❞✐r❡❝t ✇❛②✳ ❯s✐♥❣ ❡❧❡♠❡♥t❛r②tr✐❣♦♥♦♠❡tr② ✭tr✐❣♦♥♦♠❡tr② ✐♥ ❛ r✐❣❤t tr✐❛♥❣❧❡✮ ✇❡ ❝❛♥ s✉❜st✐t✉t❡ t❤❡ ❛❧t✐t✉❞❡❜❡❧♦♥❣✐♥❣ t♦ ❛ ❛s

ma = b sin γ ♦r ma = c sin β.

❚❤❡r❡❢♦r❡ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ tr✐❣♦♥♦♠❡tr✐❝ ❢♦r♠✉❧❛s

A =ab sin γ

2=

ac sin β

2=

bc sinα

2

t♦ ❝♦♠♣✉t❡ t❤❡ ❛r❡❛✳ ❆♥♦t❤❡r ✇❛② ✐s ❣✐✈❡♥ ❜② ❍ér♦♥✬s ❢♦r♠✉❧❛

A =√

s(s− a)(s− b)(s− c),

✇❤❡r❡

s =a+ b+ c

2

✾✵ ❈❍❆P❚❊❘ ✹✳ ❚❘■❆◆●▲❊❙

✐s t❤❡ s♦✲❝❛❧❧❡❞ s❡♠✐♣❡r✐♠❡t❡r✳ ❚❤❡ ❛r❡❛ ♦❢ ❛ tr✐❛♥❣❧❡ ✐s ❝❧♦s❡❧② r❡❧❛t❡❞ t♦ t❤❡t❤❡ r❛❞✐✉s r ♦❢ t❤❡ ✐♥s❝r✐❜❡❞ ❝✐r❝❧❡✳ ❙✐♥❝❡ t❤❡ ❜✐s❡❝t♦rs ♦❢ t❤❡ ✐♥t❡r✐♦r❛♥❣❧❡s ❞✐✈✐❞❡ t❤❡ tr✐❛♥❣❧❡ ✐♥t♦ t❤r❡❡ ♣❛rts t❤r♦✉❣❤ t❤❡ ❝❡♥t❡r ♦❢ t❤❡ ✐♥s❝r✐❜❡❞❝✐r❝❧❡ ❛♥❞ ❡❛❝❤ ♦❢ t❤❡s❡ tr✐❛♥❣❧❡s ❤❛s ❛❧t✐t✉❞❡ r ✇❡ ❤❛✈❡ t❤❛t

A =ar

2+

br

2+

cr

2

❛♥❞✱ ❝♦♥s❡q✉❡♥t❧②✱

r =A

s, ✭✹✳✶✮

✇❤❡r❡ s ✐s t❤❡ s❡♠✐♣❡r✐♠❡t❡r✳

✹✳✽ ❊①❡r❝✐s❡s

❊①❝❡r❝✐s❡ ✹✳✽✳✶ ❚✇♦ s✐❞❡s ♦❢ ❛ tr✐❛♥❣❧❡ ❛♥❞ t❤❡ ❛♥❣❧❡ ❡♥❝❧♦s❡❞ ❜② t❤❡♠ ❛r❡❣✐✈❡♥✿ ✸✱ ✹ ❛♥❞ ✻✵ ❞❡❣r❡❡ ✐♥ ♠❡❛s✉r❡✳

• ❋✐♥❞ t❤❡ ♠✐ss✐♥❣ s✐❞❡ ❛♥❞ ❛♥❣❧❡s✳

• ❈❛❧❝✉❧❛t❡ t❤❡ ❛r❡❛ ♦❢ t❤❡ tr✐❛♥❣❧❡✳

• ❈❛❧❝✉❧❛t❡ t❤❡ r❛❞✐✉s ♦❢ t❤❡ ❝✐r❝✉♠s❝r✐❜❡❞ ❝✐r❝❧❡ ♦❢ t❤❡ tr✐❛♥❣❧❡✳

❍✐♥t✳ ❙❡❡ t❤❡ ❝❛s❡ ❙❆❙✳

❊①❝❡r❝✐s❡ ✹✳✽✳✷ ❚❤r❡❡ s✐❞❡s ♦❢ ❛ tr✐❛♥❣❧❡ ❆❇❈ ❛r❡ ❣✐✈❡♥✿ ✻✱ ✽ ❛♥❞ ✶✷✳

• ■s ✐t ❛♥ ❛❝✉t❡✱ r✐❣❤t ♦r ♦❜t✉s❡ tr✐❛♥❣❧❡❄

• ❈❛❧❝✉❧❛t❡ t❤❡ ❛r❡❛ ♦❢ t❤❡ tr✐❛♥❣❧❡✳

• ❈❛❧❝✉❧❛t❡ t❤❡ r❛❞✐✉s ♦❢ t❤❡ ❝✐r❝✉♠s❝r✐❜❡❞ ❝✐r❝❧❡ ♦❢ t❤❡ tr✐❛♥❣❧❡✳

❍✐♥t✳ ❙❡❡ t❤❡ ❝❛s❡ ❙❙❙✳ ❚♦ ❞❡❝✐❞❡ ✇❤❡t❤❡r ❆❇❈ ✐s ❛♥ ❛❝✉t❡✱ r✐❣❤t ♦r ♦❜t✉s❡tr✐❛♥❣❧❡ ✐t ✐s ❡♥♦✉❣❤ t♦ ❝♦♠♣✉t❡ t❤❡ ❛♥❣❧❡ ♦♣♣♦s✐t❡ t♦ t❤❡ ❧♦♥❣❡st s✐❞❡ ♦❢❧❡♥❣t❤ ✶✷✿

122 = 62 + 82 − 2 · 6 · 8 · cos γ ⇒ cos γ =62 + 82 − 122

2 · 6 · 8 < 0

✇❤✐❝❤ ♠❡❛♥s t❤❛t ✇❡ ❤❛✈❡ ❛♥ ♦❜t✉s❡ ❛♥❣❧❡✳

❊①❝❡r❝✐s❡ ✹✳✽✳✸ ❚❤r❡❡ s✐❞❡s ♦❢ ❛ tr✐❛♥❣❧❡ ❛r❡ ❣✐✈❡♥✿ ✽✱ ✶✵ ❛♥❞ ✶✷✳

✹✳✽✳ ❊❳❊❘❈■❙❊❙ ✾✶

• ❈❛❧❝✉❧❛t❡ t❤❡ ❤❡✐❣❤ts ❛♥❞ t❤❡ ❛r❡❛ ♦❢ t❤❡ tr✐❛♥❣❧❡✳

• ❈❛❧❝✉❧❛t❡ t❤❡ ❜✐❣❣❡st ❛♥❣❧❡ ♦❢ t❤❡ tr✐❛♥❣❧❡✳

• ❈❛❧❝✉❧❛t❡ t❤❡ r❛❞✐✉s ♦❢ t❤❡ ❝✐r❝✉♠s❝r✐❜❡❞ ❝✐r❝❧❡ ♦❢ t❤❡ tr✐❛♥❣❧❡✳

❊①❝❡r❝✐s❡ ✹✳✽✳✹ ❚❤❡ s✐❞❡s ♦❢ ❛ tr✐❛♥❣❧❡ ❛r❡ ❛❂✺✱ ❜❂✶✷ ❛♥❞ ❝❂✶✸✳ ❈❛❧❝✉❧❛t❡t❤❡ ❛♥❣❧❡ ♦♣♣♦s✐t❡ t♦ t❤❡ s✐❞❡ ❝✳

❊①❝❡r❝✐s❡ ✹✳✽✳✺ ❚❤r❡❡ s✐❞❡s ♦❢ ❛ tr✐❛♥❣❧❡ ❛r❡ ❣✐✈❡♥✿ ✸✱ ✹ ❛♥❞√13✳

• ❋✐♥❞ t❤❡ ❛♥❣❧❡s ♦❢ t❤❡ tr✐❛♥❣❧❡✳

• ❈❛❧❝✉❧❛t❡ t❤❡ ❛r❡❛ ♦❢ t❤❡ tr✐❛♥❣❧❡✳

• ❈❛❧❝✉❧❛t❡ t❤❡ r❛❞✐✉s ♦❢ t❤❡ ❝✐r❝✉♠s❝r✐❜❡❞ ❝✐r❝❧❡ ♦❢ t❤❡ tr✐❛♥❣❧❡✳

❊①❝❡r❝✐s❡ ✹✳✽✳✻ ❚✇♦ s✐❞❡s ♦❢ ❛ tr✐❛♥❣❧❡ ❛r❡ ❛❂✽ ❛♥❞ ❜❂✻✱ t❤❡ ❛♥❣❧❡ α ♦♣♣♦✲s✐t❡ t♦ t❤❡ s✐❞❡ ❛ ✐s ✹✺ ❞❡❣r❡❡ ✐♥ ♠❡❛s✉r❡✳ ❈❛❧❝✉❧❛t❡ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ♠✐ss✐♥❣s✐❞❡ ❛♥❞ ✜♥❞ t❤❡ ❛r❡❛ ♦❢ t❤❡ tr✐❛♥❣❧❡✳

❍✐♥t✳ ❙❡❡ t❤❡ ❝❛s❡ ❙s❆✳

❊①❝❡r❝✐s❡ ✹✳✽✳✼ ❚✇♦ s✐❞❡s ♦❢ ❛ tr✐❛♥❣❧❡ ❛r❡ ❛❂✽ ❛♥❞ ❜❂✻✱ t❤❡ ❛♥❣❧❡ β ♦♣♣♦✲s✐t❡ t♦ t❤❡ s✐❞❡ ❜ ✐s ✹✺ ❞❡❣r❡❡ ✐♥ ♠❡❛s✉r❡✳ ❈❛❧❝✉❧❛t❡ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ♠✐ss✐♥❣s✐❞❡ ❛♥❞ ✜♥❞ t❤❡ ❛r❡❛ ♦❢ t❤❡ tr✐❛♥❣❧❡✳

❊①❝❡r❝✐s❡ ✹✳✽✳✽ ❋✐♥❞ t❤❡ ♠✐ss✐♥❣ q✉❛♥t✐t✐❡s ✐♥ ❡❛❝❤ r♦✇ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣t❛❜❧❡✳

❛ ❜ ❝ α β γ ❆r❡❛ ❘ r✶✷ ✷✵ 40◦

✶✷ 60◦ 40◦

✷✵ 110◦ 40◦

✶✸✳✹ ✶✽✳✺ 110◦

✷✹ ✷✺ ✸✵✶✾ ✶✷ ✾✽ ✶✵ ✷✵

✷✵ ✷✺ 60◦

✽ ✶✵ ✹✵✽ ✶✵ ✺

75◦ 25◦ 80◦ ✶

❲❛r♥✐♥❣✳ ❖❜s❡r✈❡ t❤❛t t❤❡ ❝♦s✐♥❡ r✉❧❡ ❣✐✈❡s ✐♠♣♦ss✐❜❧❡ ✈❛❧✉❡s ✐♥ ❝❛s❡ ♦❢ ❛❂✽✱❜❂✶✵ ❛♥❞ ❝❂✷✵ ✭❝❢✳ tr✐❛♥❣❧❡ ✐♥❡q✉❛❧✐t✐❡s✮✳

✾✷ ❈❍❆P❚❊❘ ✹✳ ❚❘■❆◆●▲❊❙

❊①❝❡r❝✐s❡ ✹✳✽✳✾ Pr♦✈❡ ❍ér♦♥✬ s ❢♦r♠✉❧❛✳

❍✐♥t✳ ❊①♣r❡ss t❤❡ ❝♦s✐♥❡ ♦❢ t❤❡ ❛♥❣❧❡ γ ❢r♦♠ t❤❡ ❝♦s✐♥❡ r✉❧❡✿

cos γ =a2 + b2 − c2

2ab.

❈♦♥❝❧✉❞❡ t❤❛t

sin γ =√

1− cos2 γ =

√

1− (a2 + b2 − c2)2

4a2b2.

❯s❡ t❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢♦r♠✉❧❛ t♦ ❡①♣r❡ss t❤❡ ❛r❡❛ ♦♥❧② ✐♥ t❡r♠s ♦❢ t❤❡ s✐❞❡s♦❢ t❤❡ tr✐❛♥❣❧❡✿

A =ab sin γ

2=

ab

2

√

1− (a2 + b2 − c2)2

4a2b2.

❊①❝❡r❝✐s❡ ✹✳✽✳✶✵ Pr♦✈❡ t❤❛t ✐❢ ❛ ♣♦❧②❣♦♥❛❧ s❤❛♣❡ ❤❛s ❛♥ ✐♥s❝r✐❜❡❞ ❝✐r❝❧❡t❤❡♥ t❤❡ r❛❞✐✉s ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s t❤❡ ❢r❛❝t✐♦♥ ❆✴s✱ ✇❤❡r❡ ❆ ✐s t❤❡ ❛r❡❛ ♦❢t❤❡ ♣♦❧②❣♦♥❛❧ s❤❛♣❡ ❛♥❞ s ✐s t❤❡ ❤❛❧❢ ♦❢ ✐ts ♣❡r✐♠❡t❡r✳

❈❤❛♣t❡r ✺

❊①❡r❝✐s❡s

✺✳✶ ❊①❡r❝✐s❡s

❊①❝❡r❝✐s❡ ✺✳✶✳✶ ❚❤❡ r❛❞✐✉s ♦❢ t❤❡ ❝✐r❝✉♠s❝r✐❜❡❞ ❝✐r❝❧❡ ❛r♦✉♥❞ ❛ r✐❣❤t tr✐✲❛♥❣❧❡ ✐s ✺✱ ♦♥❡ ♦❢ t❤❡ ❧❡❣s ✐s ✻✳ ❲❤❛t ✐s t❤❡ ❛r❡❛ ♦❢ t❤❡ tr✐❛♥❣❧❡❄

❙♦❧✉t✐♦♥✳ ❚❤❛❧❡s t❤❡♦r❡♠ s❛②s t❤❛t t❤❡ ❤②♣♦t❤❡♥✉s❡ ✐s ❥✉st ❝ ❂ 2 · 5 ❂ ✶✵✳❚❤❡r❡❢♦r❡ t❤❡ ♠✐ss✐♥❣ ❧❡❣ ♠✉st s❛t✐s❢② t❤❡ ❡q✉❛t✐♦♥

x2 + 62 = 102

✇❤✐❝❤ ♠❡❛♥s t❤❛t ① ❂ ✽✳ ❚❤❡ ❛r❡❛ ✐s

A =6 · 82

= 24.

❊①❝❡r❝✐s❡ ✺✳✶✳✷ ❚❤❡ ❧❡❣s ♦❢ ❛ r✐❣❤t tr✐❛♥❣❧❡ ❛r❡ ✻ ❛♥❞ ✽✳ ❍♦✇ ♠✉❝❤ ✐s t❤❡❛♥❣❧❡ ♦❢ t❤❡ ♠❡❞✐❛♥s ❜❡❧♦♥❣✐♥❣ t♦ t❤❡ ❧❡❣s❄

❋✐❣✉r❡ ✺✳✶✿ ❊①❡r❝✐s❡ ✺✳✶✳✷

✾✸

✾✹ ❈❍❆P❚❊❘ ✺✳ ❊❳❊❘❈■❙❊❙

❋✐❣✉r❡ ✺✳✷✿ ❊①❡r❝✐s❡ ✺✳✶✳✸

❙♦❧✉t✐♦♥✳ ▲❡t ❆❈ ❂ ✻ ❛♥❞ ❇❈ ❂ ✽✳ ❆t ✜rst ✇❡ ❝♦♠♣✉t❡ t❤❡ ❧❡♥❣t❤s ♦❢ t❤❡♠❡❞✐❛♥s ❜② P②t❤❛❣♦r❡❛♥ t❤❡♦r❡♠✿

32 + 82 = BF 2 ❛♥❞ 62 + 42 = AG2.

❚❤❡r❡❢♦r❡ ❇❋ ❂√73 ❛♥❞ ❆● ❂

√52✳ ■t ✐s ❦♥♦✇♥ t❤❛t t❤❡ ♠❡❞✐❛♥s ✐♥t❡rs❡❝t

❡❛❝❤ ♦t❤❡r ✉♥❞❡r t❤❡ r❛t✐♦ ✶ ✿ ✷✳ ❚❤❡r❡❢♦r❡ ✇❡ ❤❛✈❡ ❛ tr✐❛♥❣❧❡ ❝♦♥st✐t✉t❡❞ ❜②

• t❤❡ ♠✐❞❧✐♥❡ ❋● ♣❛r❛❧❧❡❧ t♦ t❤❡ ❤②♣♦t❤❡♥✉s❡ ❆❇❂✶✵ ✭❢r♦♠ t❤❡ P②t❤❛❣♦✲r❡❛♥ t❤❡♦r❡♠✮

• ✭✶✴✸✮ ❇❋ ❛♥❞ ✭✶✴✸✮ ❆●✳

❯s✐♥❣ t❤❡ ❝♦s✐♥❡ r✉❧❡ ✐t ❢♦❧❧♦✇s t❤❛t t❤❡ ❛♥❣❧❡ ω ❡♥❝❧♦s❡❞ ❜② t❤❡ ♠❡❞✐❛♥ss❛t✐s✜❡s t❤❡ ❡q✉❛t✐♦♥

52 =

(

BF

3

)2

+

(

AG

3

)2

− 2

(

BF

3

)(

AG

3

)

cosω.

❊①♣❧✐❝✐t❧②

cosω =73 + 52− 225

2√73 · 52

= − 50√3796

≈ −0.811.

❚❤❡r❡❢♦r❡ ω ≈ 144.194◦✳ ❯s✉❛❧❧② ✇❡ ❝♦♥s✐❞❡r t❤❡ ❛❝✉t❡ ❛♥❣❧❡ ✶✽✵ ✲ ω ❛s t❤❡❛♥❣❧❡ ♦❢ ♠❡❞✐❛♥s✳

❊①❝❡r❝✐s❡ ✺✳✶✳✸ ❚❤❡ ❧❡♥❣t❤s ♦❢ t❤❡ ♠❡❞✐❛♥s ♦❢ ❛♥ ✐s♦s❝❡❧❡s tr✐❛♥❣❧❡ ❛r❡ ✾✵✱✺✶ ❛♥❞ ✺✶✳ ❲❤❛t ✐s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ s✐❞❡s✱ ❛♥❞ t❤❡ ♠❡❛s✉r❡ ♦❢ t❤❡ ❛♥❣❧❡s ♦❢t❤❡ tr✐❛♥❣❧❡❄

❙♦❧✉t✐♦♥✳ ▲❡t ❆❇ ❜❡ t❤❡ s✐❞❡ ❜❡❧♦♥❣✐♥❣ t♦ t❤❡ ❧♦♥❣❡st ♠❡❞✐❛♥✳ ❙✐♥❝❡ t❤❡♠❡❞✐❛♥s ✐♥t❡rs❡❝t ❡❛❝❤ ♦t❤❡r ❜② t❤❡ r❛t✐♦ ✶ ✿ ✷ ✇❡ ❤❛✈❡ ❛ r✐❣❤t tr✐❛♥❣❧❡ t♦❝♦♠♣✉t❡ ❆❇✴✷ ❜❡❝❛✉s❡

(

AB

2

)2

+ 302 = 342.

✺✳✶✳ ❊❳❊❘❈■❙❊❙ ✾✺

❚❤❡r❡❢♦r❡ ❆❇✴✷ ❂ ✶✻✳ ❙❡❝♦♥❞❧② t❤❡ ❝♦♠♠♦♥ ❧❡♥❣t❤ ♦❢ t❤❡ ♠✐ss✐♥❣ s✐❞❡s ❝❛♥❜❡ ❝♦♠♣✉t❡❞ ❜② P②t❤❛❣♦r❡❛♥ t❤❡♦r❡♠ ❛❣❛✐♥✿

CB2 = 162 + 902 = 8356 ⇒ CB = 2√2089.

❚♦ ❝♦♠♣✉t❡ t❤❡ ❛♥❣❧❡s ✇❡ ❝❛♥ ✉s❡ ❡❧❡♠❡♥t❛r② tr✐❣♦♥♦♠❡tr② ✐♥ r✐❣❤t tr✐❛♥❣❧❡s✳❋♦r ❡①❛♠♣❧❡

tanγ

2=

BD

CD=

8

90

❛♥❞ t❤❡ ❝♦♠♠♦♥ ♠❡❛s✉r❡ ♦❢ t❤❡ ♠✐ss✐♥❣ ❛♥❣❧❡s ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❛s

α = β =180− γ

2.

❊①❝❡r❝✐s❡ ✺✳✶✳✹ ❖♥❡ ♦❢ t❤❡ ❛♥❣❧❡ ♦❢ ❛♥ ✐s♦s❝❡❧❡s tr✐❛♥❣❧❡ ✐s ✶✷✵ ❞❡❣r❡❡✱ t❤❡r❛❞✐✉s ♦❢ t❤❡ ✐♥s❝r✐❜❡❞ ❝✐r❝❧❡ ✐s ✸✳ ❍♦✇ ❧♦♥❣ ❛r❡ t❤❡ s✐❞❡s ♦❢ t❤❡ tr✐❛♥❣❧❡❄

❙♦❧✉t✐♦♥✳ ❚♦ s♦❧✈❡ t❤❡ ♣r♦❜❧❡♠ ✇❡ ✉s❡ t❤❡ ❜❛s✐❝ ❝❛s❡s ♦❢ s✐♠✐❧❛r✐t②✳ ■t ✐s❝❧❡❛r t❤❛t ❛♥ ♦❜t✉s❡ ❛♥❣❧❡ ✭❧✐❦❡ ✶✷✵✮ ❝❛♥ ♥♦t ❜❡ r❡♣❡❛t❡❞ ✐♥s✐❞❡ ❛ tr✐❛♥❣❧❡✇❤✐❝❤ ♠❡❛♥s t❤❛t t❤❡ ♠✐ss✐♥❣ ❛♥❣❧❡s ♠✉st ❜❡ ❡q✉❛❧ t♦ ❡❛❝❤ ♦t❤❡r✳ ❚❤❡②❛r❡ ✸✵ ❞❡❣r❡❡ ✐♥ ♠❡❛s✉r❡✳ ❙✐♥❝❡ t❤❡ ❛♥❣❧❡s ❛r❡ ❣✐✈❡♥ t❤❡ tr✐❛♥❣❧❡ ❆❇❈ ✐s❞❡t❡r♠✐♥❡❞ ✉♣ t♦ s✐♠✐❧❛r✐t②✳ ❲❡ ❝❛♥ ❝❤♦♦s❡ ♦♥❡ ♦❢ t❤❡ s✐❞❡ ❛r❜✐tr❛r✐❧②✿ ❧❡t✭❢♦r ❡①❛♠♣❧❡✮ t❤❡ s✐❞❡ ❆❇ ✇❤❡r❡ t❤❡ ❡q✉❛❧ ❛♥❣❧❡s ❛r❡ ❧②✐♥❣ ♦♥ ✐s ♦❢ ❧❡♥❣t❤ ✷✳❚❤❡ ❝♦♠♠♦♥ ❧❡♥❣t❤ ① ♦❢ t❤❡ ♠✐ss✐♥❣ s✐❞❡s ❝❛♥ ❜❡ ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ❝♦s✐♥❡r✉❧❡

22 = x2 + x2 − 2 · x · x · cos 120✐✳❡✳ ① ❂ ✷✴

√3✳ ◆♦✇ ✇❡ ❝❛♥ ❝♦♠♣✉t❡ t❤❡ r❛❞✐✉s ♦❢ t❤❡ ✐♥s❝r✐❜❡❞ ❝✐r❝❧❡ ❜② t❤❡

❢♦r♠✉❧❛

r =A

s,

✇❤❡r❡

A =(2/

√3)(2/

√3) sin 120

2=

1√3

✐s t❤❡ ❛r❡❛ ❛♥❞

s =AB + AC + CB

2=

2 + x+ x

2= 1 + x = 1 +

2√3≈ 2.1547

✐s t❤❡ s❡♠✐♣❡r✐♠❡t❡r ✭t❤❡ ❤❛❧❢ ♦❢ t❤❡ ♣❡r✐♠❡t❡r ♦❢ t❤❡ tr✐❛♥❣❧❡✮✳ ❋✐♥❛❧❧② t❤❡r❛t✐♦ ♦❢ t❤❡ s✐♠✐❧❛r✐t② ✐s ❥✉st r ✿ ✸ ✇❤✐❝❤ ♠❡❛♥s t❤❛t t❤❡ r❡❛❧ s✐③❡ ♦❢ t❤❡tr✐❛♥❣❧❡ ❆❇❈ ✐s

2 : AB =r

3❛♥❞ (2/

√3) : AC = (2/

√3) : BC =

r

3.

✾✻ ❈❍❆P❚❊❘ ✺✳ ❊❳❊❘❈■❙❊❙

❊①❝❡r❝✐s❡ ✺✳✶✳✺ ❖♥❡ ♦❢ t❤❡ ❛♥❣❧❡ ♦❢ ❛ tr✐❛♥❣❧❡ ✐s ✶✷✵ ❞❡❣r❡❡✱ ♦♥❡ ♦❢ t❤❡s✐❞❡s ✐s ❥✉st t❤❡ ❛r✐t❤♠❡t✐❝ ♠❡❛♥ ♦❢ t❤❡ ♦t❤❡rs✳ ❲❤❛t ✐s t❤❡ r❛t✐♦ ♦❢ t❤❡ s✐❞❡s✳

❙♦❧✉t✐♦♥✳ ❙✉♣♣♦s❡ t❤❛t a ≤ b ≤ c✳ ❚❤❡♥ ✇❡ ❤❛✈❡ t♦ ✇r✐t❡ t❤❛t

b =a+ c

2,

✐✳❡✳2 =

a

b+

c

b.

❖♥ t❤❡ ♦t❤❡r ❤❛♥❞ ❝ ♠✉st ❜❡ ♦♣♣♦s✐t❡ t♦ t❤❡ ❛♥❣❧❡ ♦❢ ♠❡❛s✉r❡ ✶✷✵ ❞❡❣r❡❡✳❯s✐♥❣ t❤❡ ❝♦s✐♥❡ r✉❧❡

c2 = a2 + b2 − 2ab cos 120.

❚❤❡r❡❢♦r❡(c

b

)2

=(a

b

)2

+ 1 +a

b

❜❡❝❛✉s❡ ♦❢ ❝♦s ✶✷✵ ❂ ✲ ✶✴✷✳ ❲❡ ❤❛✈❡ t✇♦ ❡q✉❛t✐♦♥s ✇✐t❤ t✇♦ ✉♥❦♥♦✇♥♣❛r❛♠❡t❡rs ① ❂ ❛✴❜ ❛♥❞ ② ❂ ❝✴❜ ✿

2 = x+ y ❛♥❞

y2 = x2 + 1 + x.

❚❤❡r❡❢♦r❡(2− x)2 = x2 + 1 + x,

3 = 5x ⇒ x =3

5❛♥❞ y =

7

5.

❊①❝❡r❝✐s❡ ✺✳✶✳✻ ❚❤❡ s✐❞❡s ♦❢ ❛ tr✐❛♥❣❧❡ ❤❛✈❡ ❧❡♥❣t❤s ❆❈ ❂ ❇❈ ❂√3 ❛♥❞

❆❇❂✸✳

• ❉❡t❡r♠✐♥❡ t❤❡ ❛♥❣❧❡s ❛♥❞ t❤❡ ❛r❡❛ ♦❢ t❤❡ tr✐❛♥❣❧❡✳

• ❲❤❛t ✐s t❤❡ r❛❞✐✉s ♦❢ t❤❡ ✐♥s❝r✐❜❡❞ ❝✐r❝❧❡✳

❙♦❧✉t✐♦♥✳ ❙✐♥❝❡ ✐t ✐s ❛♥ ✐s♦s❝❡❧❡s tr✐❛♥❣❧❡ t❤❡ ❝♦♠♠♦♥ ♠❡❛s✉r❡ ♦❢ t❤❡ ❛♥❣❧❡s❧②✐♥❣ ♦♥ t❤❡ s✐❞❡ ❆❇ ❝❛♥ ❜❡ ❡❛s✐❧② ❝♦♠♣✉t❡❞ ❜② ❡❧❡♠❡♥t❛r② tr✐❣♦♥♦♠❡tr②✳ ■❢❉ ✐s t❤❡ ♠✐❞♣♦✐♥t ♦❢ t❤❡ s❡❣♠❡♥t ❆❇ t❤❡♥

cosα =AD

AC=

√3

2.

✺✳✶✳ ❊❳❊❘❈■❙❊❙ ✾✼

❋✐❣✉r❡ ✺✳✸✿ ❊①❡r❝✐s❡ ✺✳✶✳✻

❚❤❡r❡❢♦r❡ α = β = 30 ❛♥❞ γ = 120✳ ❚❤❡ ❛r❡❛ ✐s

A =

√3√3 sin 120

2=

3√3

4.

❚♦ ❝♦♠♣✉t❡ t❤❡ r❛❞✐✉s ♦❢ t❤❡ ✐♥s❝r✐❜❡❞ ❝✐r❝❧❡ ✇❡ ♥❡❡❞ t❤❡ r❛t✐♦ ♦❢ t❤❡ ❛r❡❛❛♥❞ t❤❡ s❡♠✐♣❡r✐♠❡t❡r

s =AC +BC + AB

2=

2√3 + 3

2.

❋✐♥❛❧❧②

r =A

s=

3√3

2(2√3 + 3)

.

❊①❝❡r❝✐s❡ ✺✳✶✳✼ ❈❛❧❝✉❧❛t❡ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ s✐❞❡s ♦❢ ❛♥ ❡q✉✐❧❛t❡r❛❧ tr✐❛♥❣❧❡✐♥s❝r✐❜❡❞ ✐♥ ❛ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s ✶✵✳ ❈❛❧❝✉❧❛t❡ t❤❡ ❛r❡❛ ♦❢ t❤✐s tr✐❛♥❣❧❡ ❛♥❞ t❤❡r❛t✐♦ ♦❢ t❤❡ ❛r❡❛s ♦❢ t❤❡ tr✐❛♥❣❧❡ ❛♥❞ t❤❡ ❝✐r❝❧❡✳

❍✐♥t✳ ❙❡❡ ❡①❝❡r❝✐s❡ ✹✳✹✳✹✳

❊①❝❡r❝✐s❡ ✺✳✶✳✽ ❚✇♦ s✐❞❡s ♦❢ ❛ tr✐❛♥❣❧❡ ❛r❡ ❛❂✻ ❛♥❞ ❜❂✸✱ t❤❡ ❛♥❣❧❡ α ♦♣✲♣♦s✐t❡ t♦ t❤❡ s✐❞❡ ❛ ✐s ✻✵ ❞❡❣r❡❡ ✐♥ ♠❡❛s✉r❡✳ ❈❛❧❝✉❧❛t❡ t❤❡ ♠✐ss✐♥❣ s✐❞❡ ❛♥❞❛♥❣❧❡s✳ ❋✐♥❞ t❤❡ ❛r❡❛ ♦❢ t❤❡ tr✐❛♥❣❧❡✳

❙♦❧✉t✐♦♥✳ ❚❤❡ ✜rst st❡♣ ✐s t♦ ❝♦♠♣✉t❡ t❤❡ ♠✐ss✐♥❣ s✐❞❡ ❜② t❤❡ ❤❡❧♣ ♦❢ t❤❡❝♦s✐♥❡ r✉❧❡✿

62 = 32 + c2 − 2 · 3 · c · cos 60,

0 = c2 − 3c− 27.

❚❤❡r❡❢♦r❡

c12 =3±

√9 + 4 · 272

=3±

√117

2=

3± 3√13

2.

✾✽ ❈❍❆P❚❊❘ ✺✳ ❊❳❊❘❈■❙❊❙

❚❤❡ ♦♥❧② ♣♦ss✐❜❧❡ ❝❤♦✐❝❡ ✐s

c =3 + 3

√13

2≈ 6.91.

❚❤✐s ❣✐✈❡s t❤❡ ❛r❡❛ ✐♠♠❡❞✐❛t❡❧② ❜② t❤❡ ❢♦♠✉❧❛

A =3 · 3+3

√13

2· sin 60

2≈ 8.98.

❖♥❡ ♦❢ t❤❡ ♠✐ss✐♥❣ ❛♥❣❧❡ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❜② t❤❡ ❤❡❧♣ ♦❢ t❤❡ ❝♦s✐♥❡ r✉❧❡❛❣❛✐♥✿

c2 = 62 + 32 − 2 · 6 · 3 · cos γ ⇒ γ ≈ 94.42.

❋✐♥❛❧❧② β = 180− α− γ✳

❊①❝❡r❝✐s❡ ✺✳✶✳✾ ❚❤❡ ❛r❡❛ ♦❢ ❛ r✐❣❤t tr✐❛♥❣❧❡ ✐s ✸✵✱ t❤❡ s✉♠ ♦❢ t❤❡ ❧❡❣s ✐s ✶✼✳❈❛❧❝✉❧❛t❡ t❤❡ s✐❞❡s ♦❢ t❤❡ tr✐❛♥❣❧❡✳

❙♦❧✉t✐♦♥✳ ❙✐♥❝❡ ❛❜✴✷❂✸✵ ❛♥❞ ❛✰❜❂✶✼

60 = ab = a(17− a)

✇❤✐❝❤ r❡s✉❧ts ✐♥ ❛ q✉❛❞r❛t✐❝ ❡q✉❛t✐♦♥

a2 − 17a+ 60 = 0

❢♦r t❤❡ ✉♥❦♥♦✇♥ ❧❡♥❣t❤ ❛ ♦❢ ♦♥❡ ♦❢ t❤❡ ❧❡❣s✳ ❲❡ ❤❛✈❡

a12 =17±

√172 − 4 · 602

=17±

√49

2= 12 ♦r 5.

■❢ ❛ ❂ ✶✷ t❤❡♥ ❜ ❂ ✺ ❛♥❞ ✐❢ ❛ ❂ ✺ t❤❡♥ ❜ ❂ ✶✷✳

❊①❝❡r❝✐s❡ ✺✳✶✳✶✵ ❈❛❧❝✉❧❛t❡ t❤❡ ❛r❡❛ ♦❢ t❤❡ ❜r✐❣❤t ♣❛rt ✐♥ t❤❡ ✜❣✉r❡✳

❙♦❧✉t✐♦♥✳ ❚❤❡ ❛r❡❛ ✐s t❤❡ s✉♠

A = 2 · 1 + 4 + 2

2

√3 + 3

√3

❣✐✈❡♥ ❜② t❤❡ ❛r❡❛ ♦❢ ❛ r❡❝t❛♥❣❧❡✱ ❛ tr❛♣❡③♦✐❞ ❛♥❞ t❤r❡❡ ❡q✉✐❧❛t❡r❛❧ tr✐❛♥❣❧❡s✇✐t❤ s✐❞❡s ♦❢ ❧❡♥❣t❤ ✷✳

❊①❝❡r❝✐s❡ ✺✳✶✳✶✶ ❚✇♦ s✐❞❡s ♦❢ ❛ tr✐❛♥❣❧❡ ❛r❡ ✽ ❛♥❞ ✶✺✱ ✐ts ❛r❡❛ ✐s ✹✽✳ ❍♦✇❧♦♥❣ ✐s t❤❡ t❤✐r❞ s✐❞❡❄

✺✳✶✳ ❊❳❊❘❈■❙❊❙ ✾✾

❋✐❣✉r❡ ✺✳✹✿ ❊①❡r❝✐s❡ ✺✳✶✳✶✵

❙♦❧✉t✐♦♥✳ ❯s✐♥❣ t❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢♦r♠✉❧❛ ✐t ❢♦❧❧♦✇s t❤❛t

48 =8 · 15 · sin γ

2,

✐✳❡✳ s✐♥ γ ❂✵✳✽✳ ❲❡ ❛r❡ ❣♦✐♥❣ t♦ ✉s❡ t❤❡ ❝♦s✐♥❡ r✉❧❡ t♦ ❝♦♠♣✉t❡ t❤❡ s✐❞❡ ❝♦♣♣♦s✐t❡ t♦ t❤❡ ❛♥❣❧❡ γ✳ ❲❡ ❤❛✈❡ t❤❛t

cos2 γ = 1− sin2 γ = 1− 0.64 = 0.36

❛♥❞✱ ❝♦♥s❡q✉❡♥t❧②✱ ❝♦s γ ❂ ✵✳✻ ♦r ❝♦s γ ❂ ✲ ✵✳✻ ✭❛♥ ❛❝✉t❡ ♦r ❛♥ ♦❜t✉s❡ ❛♥❣❧❡✮✳❚❤❡ ❝♦s✐♥❡ r✉❧❡ s❛②s t❤❛t t❤❡ ♣♦ss✐❜❧❡ ✈❛❧✉❡s ♦❢ t❤❡ ♠✐ss✐♥❣ s✐❞❡ ❛r❡

c1 =√82 + 152 − 2 · 8 · 15 · 0.6 = 12.04

♦rc2 =

√82 + 152 + 2 · 8 · 15 · 0.6 = 20.8

❊①❝❡r❝✐s❡ ✺✳✶✳✶✷ ❚✇♦ s✐❞❡s ♦❢ ❛ tr✐❛♥❣❧❡ ❛r❡ ✽ ❛♥❞ ✶✷✱ t❤❡ ♠❡❞✐❛♥ s❡❣♠❡♥t❜❡❧♦♥❣✐♥❣ t♦ t❤❡ t❤✐r❞ s✐❞❡ ✐s ✾✳ ❲❤❛t ✐s t❤❡ ❛r❡❛ ♦❢ t❤❡ tr✐❛♥❣❧❡❄

❙♦❧✉t✐♦♥✳ ❙✐♥❝❡82 = x2 + 92 − 2 · x · 9 · cosω

❛♥❞122 = x2 + 92 − 2 · x · 12 · cos(180− ω)

✇❡ ❤❛✈❡ t❤❛t82 + 122 = 2x2 + 2 · 92

✶✵✵ ❈❍❆P❚❊❘ ✺✳ ❊❳❊❘❈■❙❊❙

❋✐❣✉r❡ ✺✳✺✿ ❊①❡r❝✐s❡ ✺✳✶✳✶✷

❜❡❝❛✉s❡ ♦❢cosω = − cos(180− ω).

❚❤❡r❡❢♦r❡ x =√23✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞

cosω =23 + 81− 64

2 ·√23 · 9

≈ 0.46 > 0,

✇❤✐❝❤ ♠❡❛♥s t❤❛tsinω =

√1− 0.462 ≈ 0.88.

❚❤❡ ❛r❡❛ ♦❢ t❤❡ tr✐❛♥❣❧❡ ❆❇❈ ✐s ♦❜✈✐♦✉s❧② t❤❡ s✉♠ ♦❢ t❤❡ ❛r❡❛s ♦❢ tr✐❛♥❣❧❡s❆❉❈ ❛♥❞ ❈❉❇✿

A =x · 9 · sinω

2+

x · 9 · sin(180− ω)

2= 2

x · 9 · sinω2

≈ 37.98

❜❡❝❛✉s❡ ♦❢sinω = sin(180− ω).

❈❤❛♣t❡r ✻

❈❧❛ss✐❝❛❧ ♣r♦❜❧❡♠s ■

✧❚❤❡ ❣r❡❛t ❜♦♦❦ ♦❢ ◆❛t✉r❡ ❧✐❡s ❡✈❡r ♦♣❡♥ ❜❡❢♦r❡ ♦✉r ❡②❡s ❛♥❞ t❤❡ tr✉❡ ♣❤✐✲❧♦s♦♣❤② ✐s ✇r✐tt❡♥ ✐♥ ✐t ✳✳✳ ❇✉t ✇❡ ❝❛♥♥♦t r❡❛❞ ✐t ✉♥❧❡ss ✇❡ ❤❛✈❡ ✜rst ❧❡❛r♥❡❞t❤❡ ❧❛♥❣✉❛❣❡ ❛♥❞ t❤❡ ❝❤❛r❛❝t❡rs ✐♥ ✇❤✐❝❤ ✐t ✐s ✇r✐tt❡♥ ✳✳✳ ■t ✐s ✇r✐tt❡♥ ✐♥♠❛t❤❡♠❛t✐❝❛❧ ❧❛♥❣✉❛❣❡ ❛♥❞ t❤❡ ❝❤❛r❛❝t❡rs ❛r❡ tr✐❛♥❣❧❡s✱ ❝✐r❝❧❡s ❛♥❞ ♦t❤❡r❣❡♦♠❡tr✐❝ ✜❣✉r❡s✳✳✳✧ ✭●❛❧✐❧❡♦ ●❛❧✐❧❡✐✮

✻✳✶ ❚❤❡ ♣r♦❜❧❡♠ ♦❢ t❤❡ t✉♥♥❡❧

Pr♦❜❧❡♠ ❬✹❪✿ ❉✉❡ t♦ t❤❡ ✐♥❝r❡❛s✐♥❣ ♣♦♣✉❧❛t✐♦♥ ❛ ❝❡rt❛✐♥ ❝✐t② ♦❢ ❛♥❝✐❡♥t ●r❡❡❝❡❢♦✉♥❞ ✐ts ✇❛t❡r s✉♣♣❧② ✐♥s✉✣❝✐❡♥t✱ s♦ t❤❛t ✇❛t❡r ❤❛❞ t♦ ❜❡ ❝❤❛♥♥❡❧❡❞ ✐♥❢r♦♠ s♦✉r❝❡ ✐♥ t❤❡ ♥❡❛r❜② ♠♦✉♥t❛✐♥s✳ ❆♥❞ s✐♥❝❡✱ ✉♥❢♦rt✉♥❛t❡❧②✱ ❛ ❧❛r❣❡ ❤✐❧❧✐♥t❡r✈❡♥❡❞✱ t❤❡r❡ ✇❛s ♥♦ ❛❧t❡r♥❛t✐✈❡ t♦ t✉♥♥❡❧✐♥❣✳ ❲♦r❦✐♥❣ ❢r♦♠ ❜♦t❤ s✐❞❡s♦❢ t❤❡ ❤✐❧❧✱ t❤❡ t✉♥♥❡❧❡rs ♠❡t ✐♥ t❤❡ ♠✐❞❞❧❡ ❛s ♣❧❛♥♥❡❞✳ ❍♦✇ ❞✐❞ t❤❡ ♣❧❛♥♥❡rs❞❡t❡r♠✐♥❡ t❤❡ ❝♦rr❡❝t ❞✐r❡❝t✐♦♥ t♦ ❡♥s✉r❡ t❤❛t t❤❡ ❝r❡✇s ✇♦✉❧❞ ♠❡❡t❄❙♦❧✉t✐♦♥✳ ❙✐♥❝❡ t❤❡ ♣♦✐♥ts ❆ ✭❝✐t②✮ ❛♥❞ ❇ ✭s♦✉r❝❡✮ ❝❛♥♥♦t ❜❡ ❝♦♥♥❡❝t❡❞❞✐r❡❝t❧② ✇❡ ❤❛✈❡ t♦ ❝♦♥♥❡❝t t❤❡♠ ✐♥❞✐r❡❝t❧②✳ ▲❡t ❈ ❜❡ ❛ ♣♦✐♥t ❢r♦♠ ✇❤✐❝❤❜♦t❤ ❆ ❛♥❞ ❇ ❛r❡ ♦❜s❡r✈❛❜❧❡✳ ❇② ♠❡❛s✉r✐♥❣ t❤❡ ❞✐st❛♥❝❡s ❆❈✱ ❇❈ ❛♥❞ t❤❡❛♥❣❧❡ γ ✇❡ ❝❛♥ ❡❛s✐❧② ✜♥❞ t❤❡ ❛♥❣❧❡s α ❛♥❞ β ❜② t❤❡ ❤❡❧♣ ♦❢ t❤❡ ❝♦s✐♥❡ r✉❧❡✳■♥♣✉ts✿ ❈❆✱ ❈❇ ❛♥❞ γ✶✳ ❈♦♠♣✉t❡

AB =√

CA2 + CB2 − 2 · CA · CB · cos γ.✷✳ ❈♦♠♣✉t❡

cosα =AB2 + AC2 − CB2

2 · AC · AB ❛♥❞ β = 180− (α + γ).

❊①❝❡r❝✐s❡ ✻✳✶✳✶ ❋✐♥❞ t❤❡ s♦❧✉t✐♦♥ ✐❢

AC = 2 ▼✐❧❡s, BC = 3 ▼✐❧❡s ❛♥❞ γ = 53◦.

✶✵✶

✶✵✷ ❈❍❆P❚❊❘ ✻✳ ❈▲❆❙❙■❈❆▲ P❘❖❇▲❊▼❙ ■

❋✐❣✉r❡ ✻✳✶✿ ❚❤❡ ♣r♦❜❧❡♠ ♦❢ t❤❡ t✉♥♥❡❧ ✲ ♦♥❡ ♦❜s❡r✈❡r

❋✐❣✉r❡ ✻✳✷✿ ❚✇♦ ♦❜s❡r✈❡rs

❊①❝❡r❝✐s❡ ✻✳✶✳✷ ❈❛♥ ②♦✉ ❣❡♥❡r❛❧✐③❡ t❤❡ ♠❡t❤♦❞ ❜② ✉s✐♥❣ ♠♦r❡ t❤❛♥ ♦♥❡♦❜s❡r✈❡rs❄

✻✳✷ ❍♦✇ t♦ ♠❡❛s✉r❡ ❛♥ ✉♥r❡❛❝❤❛❜❧❡ ❞✐st❛♥❝❡

■♥ ♠❛♥② ♣r❛❝t✐❝❛❧ s✐t✉❛t✐♦♥s t❤❡ ❞✐r❡❝t ♠❡❛s✉r✐♥❣ ♦❢ ❞✐st❛♥❝❡s ✐s ✐♠♣♦ss✐❜❧❡❀s❡❡ ❢♦r ❡①❛♠♣❧❡ ❛str♦♥♦♠✐❝❛❧ ♠❡❛s✉r❡♠❡♥ts ♦r ♥❛✈✐❣❛t✐♦♥ ♣r♦❜❧❡♠s✳ ■♥st❡❛❞♦❢ ❞✐st❛♥❝❡s ✇❡ ❝❛♥ ♠❡❛s✉r❡ ✈✐s✐❜✐❧✐t② ❛♥❣❧❡s✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦❜❧❡♠ ✐s r❡❧❛t❡❞t♦ t❤❡ ❞❡t❡r♠✐♥❛t✐♦♥ ♦❢ ❛♥ ✉♥r❡❛❝❤❛❜❧❡ ❞✐st❛♥❝❡ ❜② ♠❡❛s✉r✐♥❣ ✈✐s✐❜✐❧✐t② ❛♥❣❧❡s❛♥❞ ❛ ❣✐✈❡♥ ❜❛s❡ ❧✐♥❡✳Pr♦❜❧❡♠✿ ▲❡t t❤❡ ❞✐st❛♥❝❡ ♦❢ t❤❡ s❡❣♠❡♥t ❆❇ ❜❡ ❣✐✈❡♥ ❛♥❞ s✉♣♣♦s❡ t❤❛t ✇❡❦♥♦✇

• t❤❡ ✈✐s✐❜✐❧✐t② ❛♥❣❧❡ α ♦❢ ❇❉ ❢r♦♠ ❆✱

• t❤❡ ✈✐s✐❜✐❧✐t② ❛♥❣❧❡ β ♦❢ ❆❈ ❢r♦♠ ❇✱

• t❤❡ ✈✐s✐❜✐❧✐t② ❛♥❣❧❡ γ ♦❢ ❈❉ ❢r♦♠ ❆✱

• t❤❡ ✈✐s✐❜✐❧✐t② ❛♥❣❧❡ δ ♦❢ ❈❉ ❢r♦♠ ❇✳

✻✳✸✳ ❍❖❲ ❋❆❘ ❆❲❆❨ ■❙ ❚❍❊ ▼❖❖◆ ✶✵✸

❋✐❣✉r❡ ✻✳✸✿ ❯♥r❡❛❝❤❛❜❧❡ ❞✐st❛♥❝❡

❍♦✇ ❝❛♥ ✇❡ ❝❛❧❝✉❧❛t❡ t❤❡ ❞✐st❛♥❝❡ ❈❉❄❙♦❧✉t✐♦♥✿ ❚❤❡ s✐♥❡ r✉❧❡ ✐♥ t❤❡ tr✐❛♥❣❧❡ ❆❇❈ s❤♦✇s t❤❛t

AB

AC=

sin(π − (α + β + γ))

sin β=

sin(α + β + γ)

sin β

❛♥❞ t❤✉s

AC =sin β

sin(α + β + γ)AB.

■♥ ❛ s✐♠✐❧❛r ✇❛②

AD

AB=

sin(β + δ)

sin(π − (α + β + δ))=

sin(β + δ)

sin(α + β + δ).

❚❤❡r❡❢♦r❡

AD =sin(β + δ)

sin(α + β + δ)AB.

❯s✐♥❣ t❤❡ ❝♦s✐♥❡ r✉❧❡ ✐♥ t❤❡ tr✐❛♥❣❧❡ ❆❉❈

CD2 = AC2 + AD2 − 2 · AC · AD · cos γ.

✻✳✸ ❍♦✇ ❢❛r ❛✇❛② ✐s t❤❡ ▼♦♦♥

Pr♦❜❧❡♠ ❬✹❪✿ ❍♦✇ ❛r❡ ✇❡ t♦ ♠❡❛s✉r❡ t❤❡ ❞✐st❛♥❝❡ ♦❢ t❤❡ ▼♦♦♥ ❢r♦♠ t❤❡❊❛rt❤❄❙♦❧✉t✐♦♥✳ ❙✐♥❝❡ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❊❛rt❤ ❛♥❞ t❤❡ ▼♦♦♥ ❝❛♥♥♦t ❜❡♠❡❛s✉r❡❞ ❞✐r❡❝t❧② ✐t ♠✉st ❜❡ ♠❡❛s✉r❡❞ ✐♥❞✐r❡❝t❧②✳ ❚❤❡ ❝❛❧❝✉❧❛t✐♦♥ ♥❡❡❞s

✶✵✹ ❈❍❆P❚❊❘ ✻✳ ❈▲❆❙❙■❈❆▲ P❘❖❇▲❊▼❙ ■

❋✐❣✉r❡ ✻✳✹✿ ❍♦✇ ❢❛r ❛✇❛② ✐s t❤❡ ▼♦♦♥

❛❝❝❡ss✐❜❧❡ ❞✐st❛♥❝❡s ❧✐❦❡ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ♦❜s❡r✈❡rs ❆ ❛♥❞ ❇ ❛❧♦♥❣t❤❡ ♣❡r✐♠❡t❡r ♦❢ t❤❡ ❊❛rt❤✳ ❚❤❡② ♠❡❛s✉r❡ s✐♠✉❧t❛♥❡♦✉s❧② t❤❡ ✐♥❝❧✐♥❛t✐♦♥❛♥❣❧❡s ♦❢ t❤❡ s❡❣♠❡♥ts ❆▼ ❛♥❞ ❇▼ t♦ t❤❡ ✈❡rt✐❝❛❧ ❧✐♥❡s ♦❢ t❤❡✐r ♣♦s✐t✐♦♥s✳ ■❢✇❡ ❦♥♦✇ t❤❡ r❛❞✐✉s ♦❢ t❤❡ ❊❛rt❤ t❤❡♥ ✇❡ ❝❛♥ ❝❛❧❝✉❧❛t❡ t❤❡ ❞✐st❛♥❝❡ ❖▼ ✐♥t❤❡ ❢♦❧❧♦✇✐♥❣ ✇❛②✳■♥♣✉ts✿ t❤❡ ❛r❝❧❡♥❣t❤ ❢r♦♠ ❆ t♦ ❇✱ α✱ β ❛♥❞ t❤❡ r❛❞✐✉s ❘ ♦❢ t❤❡ ❊❛rt❤✳✶✳ ❈♦♠♣✉t❡ t❤❡ ❝❡♥tr❛❧ ❛♥❣❧❡ θ ❜② t❤❡ ❢♦r♠✉❧❛

θ (❞❡❣r❡❡)360

=t❤❡ ❛r❝❧❡♥❣t❤ ❢r♦♠ ❆ t♦ ❇

2Rπ.

❯s✐♥❣ t❤❛t ❆❖❇ ✐s ❛♥ ✐s♦s❝❡❧❡s tr✐❛♥❣❧❡

6 OAB = 6 OBA =180− θ

2.

✷✳ ❈♦♠♣✉t❡ α′ ❛♥❞ β′ ❜② t❤❡ ❢♦r♠✉❧❛s

α′ = 180− α− 180− θ

2❛♥❞ β′ = 180− β − 180− θ

2.

✸✳ ❈♦♠♣✉t❡ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ s❡❣♠❡♥t ❆❇ ❜② ✉s✐♥❣ t❤❡ ❝♦s✐♥❡ r✉❧❡ ✐♥ t❤❡✐s♦s❝❡❧❡s tr✐❛♥❣❧❡ ❆❖❇✿

AB2 = 2R2 − 2R2 cos θ.

❋r♦♠ ♥♦✇ ♦♥ t❤❡ tr✐❛♥❣❧❡ ❆▼❇ ✐s ✉♥✐q✉❡❧② ❞❡t❡r♠✐♥❡❞ ✉♣ t♦ ❝♦♥❣r✉❡♥❝❡❜❡❝❛✉s❡ ✇❡ ❦♥♦✇ ♦♥❡ s✐❞❡ ❛♥❞ t❤❡ ❛♥❣❧❡s ❧②✐♥❣ ♦♥ t❤✐s s✐❞❡✳

✻✳✸✳ ❍❖❲ ❋❆❘ ❆❲❆❨ ■❙ ❚❍❊ ▼❖❖◆ ✶✵✺

✹✳ ❈♦♠♣✉t❡ ❆▼ ❜② ✉s✐♥❣ t❤❡ s✐♥❡ r✉❧❡ ✐♥ t❤❡ tr✐❛♥❣❧❡ ❆▼❇✳✺✳ ❈♦♠♣✉t❡ ❖▼ ❜② ✉s✐♥❣ t❤❡ ❝♦s✐♥❡ r✉❧❡ ✐♥ t❤❡ tr✐❛♥❣❧❡ ❖❆▼✳

❘❡♠❛r❦ ❖♥❡ ♦❜st❛❝❧❡ r❡♠❛✐♥s❀ t❤❡ ▼♦♦♥ ♠♦✈❡s r❡❧❛t✐✈❡❧② t♦ t❤❡ ❊❛rt❤✳ ■❢t❤❡ ♦❜s❡r✈❡rs ♠❡❛s✉r❡ t❤❡ ❛♥❣❧❡s ✐♥ ❞✐✛❡r❡♥t t✐♠❡s t❤❡♥ ✇❡ ❛r❡ ❝♦♥❢r♦♥t❡❞✇✐t❤ ❛ q✉❛❞r✐❧❛t❡r❛❧ ✐♥st❡❛❞ ♦❢ ❛ tr✐❛♥❣❧❡ ❛♥❞ t❤❡ ♠❡t❤♦❞ ❤❛s ❢❛✐❧❡❞✳ ❋♦rtr✐❛♥❣✉❧❛t✐♦♥ t❤❡ ❛♥❣❧❡s ♠✉st ❜❡ ♠❡❛s✉r❡❞ s✐♠✉❧t❛♥❡♦✉s❧②✳ ■t ✐s ❝❧❡❛r t❤❛t✐❢ t❤❡ ♦❜s❡r✈❡r ♣♦s✐t✐♦♥s ❛r❡ t♦♦ ❝❧♦s❡ t♦ ❡❛❝❤ ♦t❤❡r t❤❡♥ ❆▼ ❛♥❞ ❇▼ ❛r❡❛❧♠♦st ♣❛r❛❧❧❡❧✳ ❋♦r ❛❝❝✉r❛t❡ ♠❡❛s✉r❡s ❛❧♠♦st ♣❛r❛❧❧❡❧ ❧✐♥❡s ♠✉st ❜❡ ❛✈♦✐❞❡❞✳❇✉t ❤♦✇ ✐s t❤❡ ♠❡❛s✉r❡r ❛t ❇ t♦ ❦♥♦✇ ✇❤❡♥ t❤❡ ♠❡❛s✉r❡r ❛t ❆ ✐s ♠❡❛s✉r✐♥❣❄❚❤❡ ❛♥❝✐❡♥t ●r❡❡❦✬s ❛♥s✇❡r t♦ t❤❡ ♣r♦❜❧❡♠ ✐s ❜❛s❡❞ ♦♥ ❛ s✐♠♣❧❡ ♦❜s❡r✈❛t✐♦♥✳❙✐♥❝❡ ❜♦t❤ ♠❡❛s✉r❡rs ♦❜s❡r✈❡ t❤❡ ▼♦♦♥ t❤❡ ❜❡st ✐s t♦ ✇❛✐t ❢♦r ❛ s✐❣♥❛❧ ❜② t❤❡♦❜s❡r✈❡❞ ♦❜❥❡❝t✳ ■♥ ♦t❤❡r ✇♦r❞s ♠❡❛s✉r❡rs ❤❛❞ t♦ ✇❛✐t ❢♦r s♦♠❡ ❤❛♣♣❡♥✐♥❣♦♥ t❤❡ ▼♦♦♥ ✈✐s✐❜❧❡ ❢r♦♠ ❊❛rt❤✳ ❲❤❛t ❤❛♣♣❡♥✐♥❣❄ ❆ ❧✉♥❛r ❡❝❧✐♣s❡✳ ❚❤❡❡❝❧✐♣s❡ ♣r♦✈✐❞❡s ❢♦✉r ❞✐st✐♥❝t ❡✈❡♥ts ✇❤✐❝❤ ❛r❡ ♦❜s❡r✈❛❜❧❡ s✐♠✉❧t❛♥❡♦✉s❧② ❢r♦♠❆ ❛♥❞ ❇✿

• t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ▼♦♦♥✬s ❡♥tr② t♦ t❤❡ ❊❛rt❤✬s s❤❛❞♦✇✱

• t❤❡ ❝♦♠♣❧❡t✐♦♥ ♦❢ t❤❡ ▼♦♦♥✬s ❡♥tr② t♦ t❤❡ ❊❛rt❤✬s s❤❛❞♦✇✱

• t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ▼♦♦♥✬s ❡♠❡r❣❡♥❝❡ ❢r♦♠ t❤❡ ❊❛rt❤✬s s❤❛❞♦✇✱

• t❤❡ ❝♦♠♣❧❡t✐♦♥ ♦❢ t❤❡ ▼♦♦♥✬s ❡♠❡r❣❡♥❝❡ ❢r♦♠ t❤❡ ❊❛rt❤✬s s❤❛❞♦✇✳

✶✵✻ ❈❍❆P❚❊❘ ✻✳ ❈▲❆❙❙■❈❆▲ P❘❖❇▲❊▼❙ ■

❈❤❛♣t❡r ✼

◗✉❛❞r✐❧❛t❡r❛❧s

■♥ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡ ❣❡♦♠❡tr② q✉❛❞r✐❧❛t❡r❛❧s ♠❡❛♥ ♣♦❧②❣♦♥s ✇✐t❤ ❢♦✉r s✐❞❡s❛♥❞ ❢♦✉r ✈❡rt✐❝❡s✳ ◗✉❛❞r✐❧❛t❡r❛❧s ✭♦r ♣♦❧②❣♦♥s✮ ❛r❡ t✐♣✐❝❛❧❧② ❜✉✐❧t ❢r♦♠ tr✐❛♥✲❣❧❡s ✇❤✐❝❤ ♠❛② ❤❛✈❡ ♦♥❧② ❝♦♠♠♦♥ ✈❡rt✐❝❡s ♦r s✐❞❡s✳ ❊s♣❡❝✐❛❧❧② t❤❡ q✉❛❞r✐❧❛t✲❡r❛❧s ❛r❡ t❤❡ ✉♥✐♦♥ ♦❢ t✇♦ tr✐❛♥❣❧❡s ❤❛✈✐♥❣ ❡①❛❝t❧② ♦♥❡ ❝♦♠♠♦♥ s✐❞❡✳ ❙♦♠❡✲t✐♠❡s ♦♥❡ ❛❞♠✐ts t❤❡ ✉♥✐♦♥ ♦❢ t✇♦ tr✐❛♥❣❧❡s ✇✐t❤ ❡①❛❝t❧② ♦♥❡ ❝♦♠♠♦♥ ✈❡rt❡①t♦ ❜❡ ❛ q✉❛❞r✐❧❛t❡r❛❧ ❜✉t t❤❡s❡ s❡❧❢✲✐♥t❡rs❡❝t✐♥❣ ♦r ❝r♦ss❡❞ ❝❛s❡s ✇✐❧❧ ♥♦t❜❡ ✐♠♣♦rt❛♥t ❢♦r ✉s✳ ❲❡ r❡str✐❝t ♦✉rs❡❧✈❡s t♦ t❤❡ ❝❛s❡ ♦❢ s✐♠♣❧❡ ✭♥♦t s❡❧❢✲✐♥t❡rs❡❝t✐♥❣✮ ♣♦❧②❣♦♥s✳

✼✳✶ ●❡♥❡r❛❧ ♦❜s❡r✈❛t✐♦♥s

❚❤❡♦r❡♠ ✼✳✶✳✶ ❚❤❡ s✉♠ ♦❢ t❤❡ ✐♥t❡r✐♦r ❛♥❣❧❡s ♦❢ ❛ q✉❛❞r✐❧❛t❡r❛❧ ✐s ❥✉st ✸✻✵❞❡❣r❡❡ ✐♥ ♠❡❛s✉r❡✳

❈♦r♦❧❧❛r② ✼✳✶✳✷ ❆♥② q✉❛❞r✐❧❛t❡r❛❧ ❤❛s ❛t ♠♦st ♦♥❡ ❝♦♥❝❛✈❡ ✐♥t❡r✐♦r ❛♥❣❧❡✳◗✉❛❞r✐❧❛t❡r❛❧s ❤❛✈✐♥❣ ❝♦♥❝❛✈❡ ❛♥❣❧❡s ❛r❡ ❝❛❧❧❡❞ ❝♦♥❝❛✈❡ q✉❛❞r✐❧❛t❡r❛❧s✳ ❖t❤✲❡r✇✐s❡ t❤❡ q✉❛❞r✐❧❛t❡r❛❧ ✐s ❝♦♥✈❡①✳

■♥ ✇❤❛t ❢♦❧❧♦✇s ✇❡ s✉♠♠❛r✐③❡ s♦♠❡ t②♣❡s ♦❢ q✉❛❞r✐❧❛t❡r❛❧s✳ ❚❤❡ ♠♦st✐♠♣♦rt❛♥t s♣❡❝✐❛❧ ❝❧❛ss ✐s ❢♦r♠❡❞ ❜② ♣❛r❛❧❧❡❧♦❣r❛♠s ❜❡❝❛✉s❡ ♦❢ t❤❡✐r ❝❡♥tr❛❧r♦❧❡ ✐♥ t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ t❤❡ ❊✉❝❧✐❞❡❛♥ ❣❡♦♠❡tr②✳ ❆❢t❡r ❞❡❝❧❛r✐♥❣ t❤❡ ❛①✲✐♦♠s ♦❢ ❊✉❝❧✐❞❡❛♥ ❣❡♦♠❡tr② ✇❡ ❝❛♥ ♣r♦✈❡ ❧♦ts ♦❢ ❡q✉✐✈❛❧❡♥t ❝❤❛r❛❝t❡r✐③❛t✐♦♥❢♦r ❛ ❝♦♥✈❡① q✉❛❞r✐❧❛t❡r❛❧ t♦ ❜❡ ❛ ♣❛r❛❧❧❡❧♦❣r❛♠✳ ❙♦♠❡ ♦❢ t❤❡♠ ✐s ❝r✉❝✐❛❧ t♦♣r♦✈❡ t❤❡ ♣❛r❛❧❧❡❧ ❧✐♥❡s ✐♥t❡rs❡❝t✐♥❣ t❤❡♦r❡♠ ✸✳✻✳✽✳

✶✵✼

✶✵✽ ❈❍❆P❚❊❘ ✼✳ ◗❯❆❉❘■▲❆❚❊❘❆▲❙

❋✐❣✉r❡ ✼✳✶✿ ❈❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ♣❛r❛❧❧❡❧♦❣r❛♠s

✼✳✷ P❛r❛❧❧❡❧♦❣r❛♠s

❉❡✜♥✐t✐♦♥ ❆ ♣❛r❛❧❧❡❧♦❣r❛♠ ✐s ❛ q✉❛❞r✐❧❛t❡r❛❧ ✇✐t❤ t✇♦ ♣❛✐rs ♦❢ ♣❛r❛❧❧❡❧ s✐❞❡s✳❚❤❡ ♠♦st ✐♠♣♦rt❛♥t s♣❡❝✐❛❧ ❝❛s❡s ❛r❡

• sq✉❛r❡s ✭❛❧❧ t❤❡ s✐❞❡s ❛♥❞ ❛❧❧ t❤❡ ✐♥t❡r✐♦r ❛♥❣❧❡s ♦❢ t❤❡ ♣❛r❛❧❧❡❧♦❣r❛♠❛r❡ ❡q✉❛❧✮✱

• r❡❝t❛♥❣❧❡ ✭❛❧❧ t❤❡ ✐♥t❡r✐♦r ❛♥❣❧❡s ♦❢ t❤❡ ♣❛r❛❧❧❡❧♦❣r❛♠ ❛r❡ ❡q✉❛❧✮✱

• r❤♦♠❜✉s ✭❛❧❧ t❤❡ s✐❞❡s ♦❢ t❤❡ ♣❛r❛❧❧❡❧♦❣r❛♠ ❛r❡ ❡q✉❛❧✮✳

❚❤❡♦r❡♠ ✼✳✷✳✶ ❚❤❡ q✉❛❞r✐❧❛t❡r❛❧ ❆❇❈❉ ✐s ❛ ♣❛r❛❧❡❧❧♦❣r❛♠ ✐❢ ❛♥❞ ♦♥❧② ✐❢♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s ✐s s❛t✐s✜❡❞✳

• ❚❤❡ ♦♣♣♦s✐t❡ s✐❞❡s ❛r❡ ♦❢ ❡q✉❛❧ ❧❡♥❣t❤✳

• t❤❡ ♦♣♣♦s✐t❡ ❛♥❣❧❡s ❛r❡ ❡q✉❛❧✳

• ❖♥❡ ♦❢ t❤❡ ♣❛✐rs ♦❢ t❤❡ ♦♣♣♦s✐t❡ s✐❞❡s ❛r❡ ♦❢ ❡q✉❛❧ ❧❡♥❣t❤ ❛♥❞ ♣❛r❛❧❧❡❧✳

• ■t ✐s s②♠♠❡tr✐❝ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡ ❞✐❛❣♦♥❛❧s✳

• ❚❤❡ ❞✐❛❣♦♥❛❧s ❜✐s❡❝t ❡❛❝❤ ♦t❤❡r✳

Pr♦♦❢ ■❢ ❆❇❈❉ ✐s ❛ ♣❛r❛❧❧❡❧♦❣r❛♠ t❤❡♥ ❆❙❆ ✐♠♣❧✐❡s t❤❛t ❛♥② ❞✐❛❣♦♥❛❧ ❞✐✲✈✐❞❡s t❤❡ ♣❛r❛❧❧❡❧♦❣r❛♠ ✐♥t♦ ❝♦♥❣r✉❡♥t tr✐❛♥❣❧❡s✳ ❚❤❡r❡❢♦r❡ ❜♦t❤ t❤❡ ♦♣♣♦✲s✐t❡ s✐❞❡s ❛♥❞ t❤❡ ♦♣♣♦s✐t❡ ❛♥❣❧❡s ❛r❡ ❡q✉❛❧✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞ t❤❡ ❞✐❛❣♦♥❛❧s❜✐s❡❝t ❡❛❝❤ ♦t❤❡r ❜❡❝❛✉s❡ t❤❡② ❞✐✈✐❞❡ t❤❡ ♣❛r❛❧❧❡❧♦❣r❛♠ ✐♥t♦ ❢♦✉r tr✐❛♥❣❧❡s✇❤✐❝❤ ❛r❡ ♣❛✐r✇✐s❡ ❝♦♥❣r✉❡♥t✳

❚❤❡ ♣r♦♦❢s ♦❢ t❤❡ ❝♦♥✈❡rs❡ st❛t❡♠❡♥ts ❛r❡ ❛❧s♦ ❜❛s❡❞ ♦♥ t❤❡ ❝❛s❡s ♦❢ ❝♦♥✲❣r✉❡♥❝❡ ♦❢ tr✐❛♥❣❧❡s ❛♥❞ t❤❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ♣❛r❛❧❧❡❧✐s♠✳ ■❢ t❤❡ ♦♣♣♦s✐t❡

✼✳✷✳ P❆❘❆▲▲❊▲❖●❘❆▼❙ ✶✵✾

❋✐❣✉r❡ ✼✳✷✿ P❛r❛❧❧❡❧ ❧✐♥❡s ✐♥t❡rs❡❝t✐♥❣ t❤❡♦r❡♠✿ t❤❡ ✜rst st❡♣

s✐❞❡s ❛r❡ ♦❢ ❡q✉❛❧ ❧❡♥❣t❤ t❤❡♥ ❙❙❙ ✐♠♣❧✐❡s t❤❛t ❛♥② ❞✐❛❣♦♥❛❧ ❞✐✈✐❞❡s t❤❡q✉❛❞r✐❧❛t❡r❛❧ ✐♥t♦ ❝♦♥❣r✉❡♥t tr✐❛♥❣❧❡s✳ ❚❤❡r❡❢♦r❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❛♥❣❧❡s❤❛✈❡ t❤❡ s❛♠❡ ♠❡❛s✉r❡✳ ■♥ t❤❡ s❡♥s❡ ♦❢ t❤❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ♣❛r❛❧❧❡❧✐s♠✇❡ ❤❛✈❡ t❤❛t t❤❡ ♦♣♣♦s✐t❡ s✐❞❡s ❛r❡ ♣❛r❛❧❧❡❧✳

❙✐♥❝❡ t❤❡ s✉♠ ♦❢ t❤❡ ✐♥t❡r✐♦r ❛♥❣❧❡s ✐s ✸✻✵ ❞❡❣r❡❡ ✐♥ ♠❡❛s✉r❡ t❤❡ ❡q✉❛❧✐t②♦❢ t❤❡ ♦♣♣♦s✐t❡ ❛♥❣❧❡s ♠❡❛♥s t❤❛t t❤❡ s✉♠ ♦❢ ❛♥❣❧❡s ❧②✐♥❣ ♦♥ t❤❡ s❛♠❡ s✐❞❡ ✐s✶✽✵ ❞❡❣r❡❡✳ ❚❤❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ♣❛r❛❧❧❡❧✐s♠ s❛②s t❤❛t t❤❡ ♦♣♣♦s✐t❡ s✐❞❡s❛r❡ ♣❛r❛❧❧❡❧✳

■❢ ♦♥❡ ♦❢ t❤❡ ♣❛✐rs ♦❢ t❤❡ ♦♣♣♦s✐t❡ s✐❞❡s ❛r❡ ♦❢ ❡q✉❛❧ ❧❡♥❣t❤ ❛♥❞ ♣❛r❛❧❧❡❧t❤❡♥ t❤❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ♣❛r❛❧❧❡❧✐s♠ ❛♥❞ ❙❆❙ ✐♠♣❧✐❡s t❤❛t ❛♥② ❞✐❛❣♦♥❛❧❞✐✈✐❞❡s t❤❡ q✉❛❞r✐❧❛t❡r❛❧ ✐♥t♦ ❝♦♥❣r✉❡♥t tr✐❛♥❣❧❡s✳ ❚❤❡ ♣r♦♦❢ ❝❛♥ ❜❡ ✜♥✐s❤❡❞❛s ❛❜♦✈❡✳

❚❤❡ ❧❛st t✇♦ st❛t❡♠❡♥ts ❛r❡ ♦❜✈✐♦✉s❧② ❡q✉✐✈❛❧❡♥t t♦ ❡❛❝❤ ♦t❤❡r✳ ❚❤❡r❡❢♦r❡✐t ✐s ❡♥♦✉❣❤ t♦ ❞✐s❝✉ss ♦♥❡ ♦❢ t❤❡♠✳ ❚❤❡ s②♠♠❡tr② ✇✐t❤ r❡s♣❡❝t t♦ t❤❡✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡ ❞✐❛❣♦♥❛❧s ♦❜✈✐♦✉s❧② ✐♠♣❧✐❡s t❤❛t t❤❡ ♦♣♣♦s✐t❡ s✐❞❡s ❛r❡♣❛r❛❧❧❡❧✳

❆s ❛♥ ❛♣♣❧✐❝❛t✐♦♥ ✇❡ ♣r♦✈❡ t❤❡ ♣❛r❛❧❧❡❧ ❧✐♥❡s ✐♥t❡rs❡❝t✐♥❣ t❤❡♦r❡♠ ✸✳✻✳✽✶st st❡♣❲❡ ❝❛♥ ❝♦♥❝❧✉❞❡ t❤❛t t❤❡ ♣❛r❛❧❧❡❧ ♣r♦❥❡❝t✐♦♥s ♦❢ ❝♦♥❣r✉❡♥t s❡❣♠❡♥ts❛r❡ ❝♦♥❣r✉❡♥t✿ ✐❢ ❖❆❂❆❇ t❤❡♥ t❤❡ tr✐❛♥❣❧❡s ❖❆❆✬ ❛♥❞ ❆❇❈ ❛r❡ ❝♦♥❣r✉❡♥t❛♥❞ ❆❈❂❆✬❇✬❜② t❤❡♦r❡♠ ✼✳✷✳✶✳ ❚❤❡r❡❢♦r❡ ❖❆✿❖❇❂❖❆✬✿❖❇✬❂✶✿✷✳✷♥❞ st❡♣ ■♥ ❝❛s❡ ♦❢ ♥♦t ♥❡❝❡ss❛r✐❧② ❝♦♥❣r✉❡♥t s❡❣♠❡♥ts ❖❆ ❛♥❞ ❆❇ ❧❡t n❜❡ ❛♥ ❛r❜✐tr❛r② ✐♥t❡❣❡r ❛♥❞ ❞✐✈✐❞❡ t❤❡ s❡❣♠❡♥t ❖❆ ✐♥t♦ ♥ ❡q✉❛❧ ♣❛rts ❜② t❤❡♣♦✐♥ts

X0 = O, X1, . . . , Xn = A.

❈♦♥t✐♥✉❡ t❤❡ ♣r♦❝❝❡ss ♦❢ ❝♦♣②✐♥❣ t❤❡ s❡❣♠❡♥t ♦❢ ❧❡♥❣t❤ ❖❆✴♥ ❢r♦♠ ❆ ✐♥t♦t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ ❇ ❛s ❢❛r ❛s ✇❡ ❤❛✈❡

kOA

n≤ OB ≤ (k + 1)

OA

n.

✶✶✵ ❈❍❆P❚❊❘ ✼✳ ◗❯❆❉❘■▲❆❚❊❘❆▲❙

❯s✐♥❣ t❤❡ ✜rst st❡♣ t❤❡ ♣❛r❛❧❧❡❧ ♣r♦❥❡❝t✐♦♥s

X ′0 = O, X ′

1, . . . , X ′n = A′

❣✐✈❡s t❤❡ ❞✐✈✐s♦♥ ♦❢ ❖❆✬ ✐♥t♦ ♥ ❡q✉❛❧ ♣❛rts✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞

kOA′

n≤ OB′ ≤ (k + 1)

OA′

n.

❚❤❡r❡❢♦r❡k

n≤ OB

OA≤ k + 1

n❛♥❞

k

n≤ OB′

OA′ ≤k + 1

n

✇❤✐❝❤ ♠❡❛♥s t❤❛t∣

∣

∣

∣

OB

OA− OB′

OA′

∣

∣

∣

∣

≤ 1

n

❢♦r ❛♥② ✐♥t❡❣❡r n ∈ N✳ ❚❛❦✐♥❣ t❤❡ ❧✐♠✐t n → ∞ ✇❡ ❤❛✈❡ t❤❛t

OB

OA=

OB′

OA′

❛s ✇❛s t♦ ❜❡ st❛t❡❞✳

✼✳✸ ❙♣❡❝✐❛❧ ❝❧❛ss❡s ♦❢ q✉❛❞r✐❧❛t❡r❛❧s

❉❡✜♥✐t✐♦♥ ❆ q✉❛❞r✐❧❛t❡r❛❧ ✐s ❝❛❧❧❡❞ tr❛♣❡③♦✐❞ ✐❢ ✐t ❤❛s ❛t ❧❡❛st ♦♥❡ ♣❛✐r♦❢ ♦♣♣♦s✐t❡ s✐❞❡s ✇❤✐❝❤ ❛r❡ ♣❛r❛❧❧❡❧✳ ❆♥ ✐s♦s❝❡❧❡s tr❛♣❡③♦✐❞ ♦r s②♠♠❡tr✐❝tr❛♣❡③♦✐❞ ❤❛✈❡ ❡q✉❛❧ ❜❛s❡ ❛♥❣❧❡s ✐♥ ♠❡❛s✉r❡✳

❉❡✜♥✐t✐♦♥ ❆ q✉❛❞r✐❧❛t❡r❛❧ ✐s ❝❛❧❧❡❞ ❦✐t❡ ✐❢ t✇♦ ♣❛✐rs ♦❢ ❛❞❥❛❝❡♥t s✐❞❡s ❛r❡ ♦❢❡q✉❛❧ ❧❡♥❣t❤✳

❊①❝❡r❝✐s❡ ✼✳✸✳✶ Pr♦✈❡ t❤❛t ✐♥ ❝❛s❡ ♦❢ ❛ ❦✐t❡ t❤❡ ❛♥❣❧❡s ❜❡t✇❡❡♥ t❤❡ t✇♦ ♣❛✐rs♦❢ ❡q✉❛❧ s✐❞❡s ❛r❡ ❡q✉❛❧ ✐♥ ♠❡❛s✉r❡ ❛♥❞ t❤❡ ❞✐❛❣♦♥❛❧s ❛r❡ ♣❡r♣❡♥❞✐❝✉❧❛r✳

❙♦❧✉t✐♦♥✳ ❋r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛ ❦✐t❡ ♦♥❡ ♦❢ t❤❡ ❞✐❛❣♦♥❛❧ ❞✐✈✐❞❡s t❤❡ ❦✐t❡✐♥t♦ ❝♦♥❣r✉❡♥t tr✐❛♥❣❧❡s ❜② t❤❡ ❜❛s✐❝ ❝❛s❡ ❙❙❙ ♦❢ t❤❡ ❝♦♥❣r✉❡♥❝❡✳ ❚❤❡ ♣❡r✲♣❡♥❞✐❝✉❧❛r✐t② ❢♦❧❧♦✇s ❞✐r❡❝t❧② ❢r♦♠ t❤❡ ❣❡♦♠❡tr✐❝ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ t❤❡ ♣❡r✲♣❡♥❞✐❝✉❧❛r ❜✐s❡❝t♦r✳

✼✳✸✳ ❙P❊❈■❆▲ ❈▲❆❙❙❊❙ ❖❋ ◗❯❆❉❘■▲❆❚❊❘❆▲❙ ✶✶✶

❋✐❣✉r❡ ✼✳✸✿ ❆①✐❛❧❧② s②♠♠❡tr✐❝ q✉❛❞r✐❧❛t❡r❛❧s

✼✳✸✳✶ ❙②♠♠❡tr✐❡s

❙✉♣♣♦s❡ t❤❛t t❤❡ q✉❛❞r✐❧❛t❡r❛❧ ❆❇❈❉ ❤❛s ❛♥ ❛①✐❛❧ s②♠♠❡tr②✱ ✐✳❡✳ ✇❡ ❤❛✈❡❛ ❧✐♥❡ s✉❝❤ t❤❛t t❤❡ q✉❛❞r✐❧❛t❡r❛❧ ✐s ✐♥✈❛r✐❛♥t ✉♥❞❡r t❤❡ r❡✢❡❝t✐♦♥ ❛❜♦✉t t❤✐s❧✐♥❡✳ ❙✐♥❝❡ ❛♥② ✈❡rt❡① ♠✉st ❜❡ tr❛♥s❢♦r♠❡❞ ✐♥t♦ ❛♥♦t❤❡r ♦♥❡ ✇❡ ❤❛✈❡ t❤❛t

k + l = 4,

✇❤❡r❡ t❤❡ ♥✉♠❜❡r ❦ ♦❢ t❤❡ ✈❡rt✐❝❡s ✇❤✐❝❤ ❛r❡ ♥♦t ♦♥ t❤❡ ❛①✐s ♦❢ s②♠♠❡tr②♠✉st ❜❡ ❡✈❡♥✳ ❚❤❡ ♣♦ss✐❜❧❡ ❝❛s❡s ❛r❡ ❦❂✵✱ ✷ ♦r ✹✿

0 + 4 = 4, 2 + 2 = 4 ❛♥❞ 4 + 0 = 4.

❚❤❡ ❝❛s❡ k = 0 ✐s ♦❜✈✐♦✉s❧② ✐♠♣♦ss✐❜❧❡✳ ■❢ ✇❡ ❤❛✈❡ ✷ ✈❡rt✐❝❡s ♦♥ t❤❡ ❛①✐s ♦❢s②♠♠❡tr② t❤❡♥ t❤❡ q✉❛❞r✐❧❛t❡r❛❧ ♠✉st ❜❡ ❛ ❝♦♥✈❡① ♦r ❝♦♥❝❛✈❡ ❦✐t❡✳ ❖t❤❡r✇✐s❡✐t ✐s ❛ s②♠♠❡tr✐❝ tr❛♣❡③✐✉♠✳

❉❡✜♥✐t✐♦♥ ❘♦t❛t✐♦♥❛❧ s②♠♠❡tr② ♦❢ ♦r❞❡r ♥ ✇✐t❤ r❡s♣❡❝t t♦ ❛ ♣❛rt✐❝✉❧❛r♣♦✐♥t ♠❡❛♥s t❤❛t r♦t❛t✐♦♥s ❜② ❛♥❣❧❡ ✸✻✵✴♥ ❞♦❡s ♥♦t ❝❤❛♥❣❡ t❤❡ ♦❜❥❡❝t✳

❊①❝❡r❝✐s❡ ✼✳✸✳✷ Pr♦✈❡ t❤❛t q✉❛❞r✐❧❛t❡r❛❧s ✇✐t❤ s②♠♠❡tr② ♦❢ ♦r❞❡r ✷ ❛r❡ ♣❛r✲❛❧❧❡❧♦❣r❛♠s

❊①❝❡r❝✐s❡ ✼✳✸✳✸ Pr♦✈❡ t❤❛t q✉❛❞r✐❧❛t❡r❛❧s ✇✐t❤ s②♠♠❡tr② ♦❢ ♦r❞❡r ✹ ❛r❡sq✉❛r❡s✳

✶✶✷ ❈❍❆P❚❊❘ ✼✳ ◗❯❆❉❘■▲❆❚❊❘❆▲❙

✼✳✸✳✷ ❆r❡❛

❚❤❡ ❛r❡❛ ♦❢ ❛ ♣♦❧②❣♦♥❛❧ r❡❣✐♦♥ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❛s t❤❡ s✉♠ ♦❢ t❤❡ ❛r❡❛s ♦❢s✉❜tr✐❛♥❣❧❡s✳ ■♥ ✇❤❛t ❢♦❧❧♦✇s ✇❡ ❝♦♥s✐❞❡r s♦♠❡ s♣❡❝✐❛❧ ❝❛s❡s ✇✐t❤ ❡①♣❧✐❝✐t❢♦r♠✉❧❛s✳ ❚❤❡② ❛r❡ ❡❛s② ❝♦♥s❡q✉❡♥❝❡s ♦❢ t❤❡ tr✐❛♥❣❧❡ ❞❡❝♦♠♣♦s✐t✐♦♥✳ ❚❤❡❛r❡❛ ♦❢ ❛

• ♣❛r❛❧❧❡❧♦❣r❛♠ ✐s t❤❡ ♣r♦❞✉❝t ♦❢ ♦♥❡ ♦❢ t❤❡ ♣❛r❛❧❧❡❧ ❜❛s❡s ❛♥❞ t❤❡ ❛❧t✐t✉❞❡❜❡❧♦♥❣✐♥❣ t♦ t❤✐s ❜❛s❡✳ ❚❤❡ tr✐❣♦♥♦♠❡tr✐❝ ✈❡rs✐♦♥ ♦❢ t❤❡ ❢♦r♠✉❧❛ ✐s

A = ab sinα.

❚❤✐s ❢♦❧❧♦✇s ❡❛s✐❧② ❢r♦♠ t❤❡ ❞✐✈✐s✐♦♥ ♦❢ t❤❡ ♣❛r❛❧❧❡❧♦❣r❛♠ ✐♥t♦ ❝♦♥❣r✉❡♥ttr✐❛♥❣❧❡s ❜② ♦♥❡ ♦❢ t❤❡ ❞✐❛❣♦♥❛❧s✳

• tr❛♣❡③♦✐❞ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❛s

A =a+ c

2m,

✇❤❡r❡ ❛ ❛♥❞ ❝ ❛r❡ t❤❡ ❧❡♥❣t❤s ♦❢ t❤❡ ♣❛r❛❧❧❡❧ ❜❛s❡s ❛♥❞ ♠ ✐s t❤❡ ❛❧t✐t✉❞❡♦❢ t❤❡ tr❛♣❡③♦✐❞✳ ❖♥❡ ❝❛♥ ✐♥tr♦❞✉❝❡ t❤❡ ♠✐❞✲❧✐♥❡ s❡❣♠❡♥t ❢♦r tr❛♣❡③♦✐❞s♦♥ t❤❡ ♠♦❞❡❧ ♦❢ tr✐❛♥❣❧❡s ✐♥ t❤❡ s❛♠❡ ✇❛②✿ t❤❡ ♠✐❞❧✐♥❡ ♦❢ ❛ tr❛♣❡③♦✐❞✐s ❥✉st t❤❡ ❧✐♥❡ s❡❣♠❡♥t ❥♦✐♥✐♥❣ t❤❡ ♠✐❞♣♦✐♥ts ♦❢ t❤❡ ❧❡❣s✳ ❯s✐♥❣ t❤❡❞✐✈✐s✐♦♥ ♦❢ t❤❡ tr❛♣❡③♦✐❞ ✐♥t♦ tr✐❛♥❣❧❡s ❜② ♦♥❡ ♦❢ t❤❡ ❞✐❛❣♦♥❛❧s ✐t ❝❛♥❜❡ ❡❛s✐❧② s❡❡♥ t❤❛t t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ♠✐❞❧✐♥❡ ♦❢ ❛ tr❛♣❡③♦✐❞ ✐s ❥✉st t❤❡❛r✐t❤♠❡t✐❝ ♠❡❛♥ ♦❢ t❤❡ ❧❡♥❣t❤s ♦❢ t❤❡ ♣❛r❛❧❧❡❧ ❜❛s❡s✳ ❆♥♦t❤❡r ✇❛② t♦❝♦♥❝❧✉❞❡ t❤❡ ❛r❡❛ ❢♦r♠✉❧❛ ✐s t♦ ♣✉t t✇♦ ❝♦♥❣r✉❡♥t ❝♦♣✐❡s ♦❢ t❤❡ tr❛♣❡✲③♦✐❞ ♥❡①t t♦ ❡❛❝❤ ♦t❤❡r ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t t❤❡② ❢♦r♠ ❛ ♣❛r❛❧❧❡❧♦❣r❛♠✳■♥ t❡r♠s ♦❢ ❣❡♦♠❡tr✐❝ tr❛♥s❢♦r♠❛t✐♦♥ ✐t ❝❛♥ ❜❡ r❡❛❧✐③❡❞ ❜② ❛ ❝❡♥tr❛❧r❡✢❡❝t✐♦♥ ❛❜♦✉t t❤❡ ♠✐❞♣♦✐♥t ♦❢ ♦♥❡ ♦❢ t❤❡ ❧❡❣s✳

• ❝♦♥✈❡① q✉❛❞r✐❧❛t❡r❛❧ ✐s ❥✉st

A =ef sinω

2,

✇❤❡r❡ ❡ ❛♥❞ ❢ ❛r❡ t❤❡ ❧❡♥❣t❤s ♦❢ t❤❡ ❞✐❛❣♦♥❛❧s ❛♥❞ ω ✐s t❤❡ ❛♥❣❧❡ ❡♥❝❧♦s❡❞❜② t❤❡♠✳

❊①❝❡r❝✐s❡ ✼✳✸✳✹ Pr♦✈❡ t❤❡ ❛r❡❛ ❢♦r♠✉❧❛ ♦❢ ❛ ♣❛r❛❧❧❡❧♦❣r❛♠✳

❊①❝❡r❝✐s❡ ✼✳✸✳✺ Pr♦✈❡ t❤❡ ❛r❡❛ ❢♦r♠✉❧❛ ♦❢ ❛ tr❛♣❡③♦✐❞✳

❊①❝❡r❝✐s❡ ✼✳✸✳✻ Pr♦✈❡ t❤❡ ❛r❡❛ ❢♦r♠✉❧❛ ♦❢ ❛ ❦✐t❡✳

✼✳✸✳ ❙P❊❈■❆▲ ❈▲❆❙❙❊❙ ❖❋ ◗❯❆❉❘■▲❆❚❊❘❆▲❙ ✶✶✸

❚❤❡♦r❡♠ ✼✳✸✳✼ ▲❡t ❆❇❈❉ ❜❡ ❛ ❝♦♥✈❡① q✉❛❞r✐❧❛t❡r❛❧✳ ❚❤❡ ❛r❡❛ ❝❛♥ ❜❡ ❝♦♠✲♣✉t❡❞ ❛s

A =AC · BD · sinω

2,

✇❤❡r❡ ω ✐s t❤❡ ❛♥❣❧❡ ❡♥❝❧♦s❡❞ ❜② t❤❡ ❞✐❛❣♦♥❛❧s ❆❈ ❛♥❞ ❇❉✳

Pr♦♦❢ ▲❡t ❊ ❜❡ t❤❡ ♣♦✐♥t ✇❤❡r❡ t❤❡ ❞✐❛❣♦♥❛❧s ♠❡❡t ❛t✳ ❚❤❡ tr✐❛♥❣❧❡s ❆❊❇✱❇❊❈✱ ❈❊❉ ❛♥❞ ❉❊❆ ❝♦✈❡rs t❤❡ q✉❛❞r✐❧❛t❡r❛❧ s✉❝❤ t❤❛t ✇❡ ❤❛✈❡ ♦♥❧② ❝♦♠♠♦♥✈❡rt✐❝❡s ❛♥❞ ❡❞❣❡s✳ ❚❤❡r❡❢♦r❡ t❤❡ ❛r❡❛ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❛s t❤❡ s✉♠

A = AAEB + ABEC + ACED + ADEA.

❙✐♥❝❡ t❤❡ ❛♥❣❧❡s ❛t t❤❡ ❝♦♠♠♦♥ ✈❡rt❡① ❊ ❛r❡ ❛❧t❡r♥❛t❡❧② ω ❛♥❞ ✶✽✵ ✲ ω ✇❡❝❛♥ ❝♦♥❝❧✉❞❡ t❤❛t

A =AE · EB + EB · EC + EC · ED + ED · EA

2sinω =

AC · BD · sinω2

,

✇❤❡r❡ ω ✐s t❤❡ ❛♥❣❧❡ ❡♥❝❧♦s❡❞ ❜② t❤❡ ❞✐❛❣♦♥❛❧s ❆❈ ❛♥❞ ❇❉✳

❊①❝❡r❝✐s❡ ✼✳✸✳✽ ▲❡t ❆❇❈❉ ❜❡ ❛ ❝♦♥✈❡① q✉❛❞r✐❧❛t❡r❛❧✳ ❋✐♥❞ t❤❡ ♣♦✐♥t ✐♥ t❤❡♣❧❛♥❡ t♦ ♠✐♥✐♠✐③❡ t❤❡ s✉♠

XA+XB +XC +XD.

❙♦❧✉t✐♦♥✳ ❇② t❤❡ tr✐❛♥❣❧❡ ✐♥❡q✉❛❧✐t② t❤❡ ♣♦✐♥t ❳ ♠✉st ❜❡ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢t❤❡ ❞✐❛❣♦♥❛❧s ❆❈ ❛♥❞ ❇❉✳

✶✶✹ ❈❍❆P❚❊❘ ✼✳ ◗❯❆❉❘■▲❆❚❊❘❆▲❙

❈❤❛♣t❡r ✽

❊①❡r❝✐s❡s

✽✳✶ ❊①❡r❝✐s❡s

❊①❝❡r❝✐s❡ ✽✳✶✳✶ ❚❤r❡❡ s✐❞❡s ♦❢ ❛ s②♠♠❡tr✐❝❛❧ tr❛♣❡③♦✐❞ ❛r❡ ♦❢ ❧❡♥❣t❤ ✶✵✳❚❤❡ ❢♦✉rt❤ s✐❞❡ ❤❛s ❧❡♥❣t❤ ✷✵✳ ❈❛❧❝✉❧❛t❡ t❤❡ ❛♥❣❧❡s ❛♥❞ t❤❡ ❛r❡❛ ♦❢ t❤❡ tr❛♣❡✲③♦✐❞✳

❙♦❧✉t✐♦♥✳ ❯s✐♥❣ t❤❡ s②♠♠❡tr② ✇❡ ❝❛♥ ❡❛s✐❧② ❝❤❛♥❣❡ t❤❡ tr❛♣❡③♦✐❞ ✐♥t♦ ❛r❡❝t❛♥❣❧❡✳ ▲❡t ❆❇❈❉ ❜❡ ❛ s②♠♠❡tr✐❝❛❧ tr❛♣❡③♦✐❞ ❤❛✈✐♥❣ s✐❞❡s ♦❢ ❧❡♥❣t❤ ❆❇❂ ✷✵✱ ❇❈ ❂ ❆❉ ❂ ✶✵ ❛♥❞ ❈❉ ❂ ✶✵✳ ❚❤❡ ♦rt❤♦❣♦♥❛❧ ♣r♦❥❡❝t✐♦♥ ❈✬❉✬ ♦❢ ❈❉♦♥t♦ t❤❡ ❧♦♥❣❡r ❜❛s❡ ❆❇ ✐s ♦❢ ❧❡♥❣t❤ ✶✵ ❛❣❛✐♥✳ ❚❤❡r❡❢♦r❡ ❆❉✬❂✺ ❛♥❞ ❇❈✬ ❂✺ ❜❡❝❛✉s❡ ♦❢ t❤❡ s②♠♠❡tr②✳ ❋r♦♠ t❤❡ r✐❣❤t tr✐❛♥❣❧❡ ❆❉✬❉ ✇❡ ❤❛✈❡ t❤❛t t❤❡❤❡✐❣❤t ✐s

DD′ =√102 − 52 =

√75.

❚❤❡ s✐❞❡s ♦❢ t❤❡ r❡❝t❛♥❣❧❡ ✐s ❥✉st ❛ ❂ ❆❇ ✲ ❇❈✬ ❂ ✷✵ ✲ ✺ ❂ ✶✺ ❛♥❞ ❜ ❂√75✳

❚❤❡ ❛r❡❛ ✐sA = 15

√75 = 75

√3.

❚❤❡ ❛♥❣❧❡s ❛r❡ ❛❧t❡r♥❛t❡❧② ✻✵ ❛♥❞ ✶✷✵ ❞❡❣r❡❡ ✐♥ ♠❡❛s✉r❡✳

❋✐❣✉r❡ ✽✳✶✿ ❊①❡r❝✐s❡ ✽✳✶✳✶

✶✶✺

✶✶✻ ❈❍❆P❚❊❘ ✽✳ ❊❳❊❘❈■❙❊❙

❋✐❣✉r❡ ✽✳✷✿ ❊①❡r❝✐s❡s ✽✳✶✳✷ ❛♥❞ ✽✳✶✳✸

❊①❝❡r❝✐s❡ ✽✳✶✳✷ ❚❤❡ s✐❞❡s ❆❇ ❛♥❞ ❇❈ ♦❢ r❡❝t❛♥❣❧❡ ❆❇❈❉ ❛r❡ ✶✵ ❛♥❞ ✻✳❲❤❛t ✐s t❤❡ ❞✐st❛♥❝❡ ♦❢ ❛ ♣♦✐♥t P ♦♥ t❤❡ s✐❞❡ ❆❇ ❢r♦♠ t❤❡ ✈❡rt❡① ❉ ✐❢

AP + PC = 12.

❙♦❧✉t✐♦♥✳ ❋r♦♠ t❤❡ r✐❣❤t tr✐❛♥❣❧❡ P❇❈

PB2 + 62 = PC2,

✐✳❡✳(10− AP )2 + 36 = PC2.

❙✐♥❝❡ ❆P✰P❈❂✶✷

(10− AP )2 + 36 = (12− AP )2

❛♥❞4 · AP = 8 ⇒ AP = 2.

❋✐♥❛❧❧②PD2 = 62 + 22 ⇒ PD =

√40.

❊①❝❡r❝✐s❡ ✽✳✶✳✸ ■♥ ❛ s②♠♠❡tr✐❝❛❧ tr❛♣❡③♦✐❞ t❤❡ ✐♥❝❧✐♥❛t✐♦♥ ❛♥❣❧❡ ♦❢ t❤❡ ❞✐✲❛❣♦♥❛❧ t♦ t❤❡ ❧♦♥❣❡r ♣❛r❛❧❧❡❧ ❜❛s❡ ✐s ✹✺ ❞❡❣r❡❡✱ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❞✐❛❣♦♥❛❧ ✐s✶✵✳ ❲❤❛t ✐s t❤❡ ❛r❡❛ ♦❢ t❤❡ tr❛♣❡③♦✐❞❄

❙♦❧✉t✐♦♥✳ ❯s✐♥❣ t❤❡ s②♠♠❡tr② ✇❡ ❝❛♥ ❡❛s✐❧② ❝❤❛♥❣❡ t❤❡ tr❛♣❡③♦✐❞ ✐♥t♦ ❛r❡❝t❛♥❣❧❡✳ ❙✐♥❝❡ t❤❡ ❞✐❛❣♦♥❛❧ ❜✐s❡❝ts t❤❡ ❛♥❣❧❡s ♦❢ t❤❡ r❡❝t❛♥❣❧❡ ✐t ♠✉st❜❡ ❛ sq✉❛r❡✳ ❚❤❡ ❝♦♠♠♦♥ ❧❡♥❣t❤ ① ♦❢ t❤❡ s✐❞❡s ❝❛♥ ❜❡ ❞❡r✐✈❡❞ ❢r♦♠ t❤❡P②t❤❛❣♦r❡❛♥ t❤❡♦r❡♠

x2 + x2 = 102 ⇒ x =√50 = 5

√2

❛♥❞ t❤❡ ❛r❡❛ ✐s x2 ❂ ✺✵✳

✽✳✶✳ ❊❳❊❘❈■❙❊❙ ✶✶✼

❋✐❣✉r❡ ✽✳✸✿ ❊①❡r❝✐s❡ ✽✳✶✳✹

❊①❝❡r❝✐s❡ ✽✳✶✳✹ ❚❤❡ s✐❞❡ ♦❢ t❤❡ sq✉❛r❡ ❆❇❈❉ ✐s ✶✵✳ ❈❛❧❝✉❧❛t❡ t❤❡ r❛❞✐✉s♦❢ t❤❡ ❝✐r❝❧❡ ✇❤✐❝❤ ♣❛ss❡s t❤r♦✉❣❤ t❤❡ ♣♦✐♥t ❆✱ ❛♥❞ t♦✉❝❤❡s t❤❡ s✐❞❡s ❇❈ ❛♥❞❈❉✳

❙♦❧✉t✐♦♥✳ ❉✐✈✐❞❡ t❤❡ ♣r♦❜❧❡♠ ✐♥t♦ t✇♦ ♣❛rts✳ ❆t ✜rst ❧❡t ✉s ❝♦♥❝❡♥tr❛t❡ ♦♥t❤❡ ❝✐r❝❧❡s t♦✉❝❤✐♥❣ t❤❡ s✐❞❡s ❇❈ ❛♥❞ ❈❉✳ ❚❤❡ ❝❡♥t❡r ♦❢ s✉❝❤ ❛ ❝✐r❝❧❡ ♠✉st❜❡ ♦♥ t❤❡ ❞✐❛❣♦♥❛❧ ❈❆ ♦❢ t❤❡ sq✉❛r❡✳ ▲❡t ① ❜❡ t❤❡ ❞✐st❛♥❝❡ ♦❢ t❤❡ ❝❡♥t❡r❢r♦♠ ❈✳ P②t❤❛❣♦r❡❛♥ t❤❡♦r❡♠ s❛②s t❤❛t

x2 = r2x + r2x, ✐✳❡✳ x = rx√2,

✇❤❡r❡ rx ✐s t❤❡ r❛❞✐✉s ♦❢ t❤❡ ❝✐r❝❧❡✳ ■t ✐s ❧❛❜❡❧❧❡❞ ❜② t❤❡ ❝♦♦r❞✐♥❛t❡ ①✳ ❚❤❡♣♦✐♥t ❆ ❤❛s ❝♦♦r❞✐♥❛t❡ xA = 10

√2✳ ❚❤❡ ❝✐r❝❧❡ ♣❛ss❡s t❤r♦✉❣❤ ❆ ✐❢ ❛♥❞ ♦♥❧②

✐❢|x− xA| = rx, ✐✳❡✳ x− 10

√2 = rx ♦r 10

√2− x = rx.

❲❡ ❤❛✈❡rx√2− 10

√2 = rx ♦r 10

√2− rx

√2 = rx.

❚❤❡r❡❢♦r❡

rx =10√2√

2− 1♦r rx =

10√2√

2 + 1.

❊①❝❡r❝✐s❡ ✽✳✶✳✺ ■♥ r❡❝t❛♥❣❧❡ ❆❇❈❉ s✐❞❡ ❆❇ ✐s t❤r❡❡ t✐♠❡s ❧♦♥❣❡r t❤❡♥ ❇❈✳❚❤❡ ❞✐st❛♥❝❡ ♦❢ ❛♥ ✐♥t❡r✐♦r ♣♦✐♥t P ❢r♦♠ t❤❡ ✈❡rt✐❝❡s ❇✱ ❆ ❛♥❞ ❉ ✐s P❇ ❂4√2✱ P❆ ❂

√2 ❛♥❞ P❉ ❂ ✷✳ ❲❤❛t ✐s t❤❡ ❛r❡❛ ♦❢ t❤❡ r❡❝t❛♥❣❧❡✳

✶✶✽ ❈❍❆P❚❊❘ ✽✳ ❊❳❊❘❈■❙❊❙

❋✐❣✉r❡ ✽✳✹✿ ❊①❡r❝✐s❡ ✽✳✶✳✻

❋♦r t❤❡ s♦❧✉t✐♦♥ s❡❡ ❊①❡r❝✐s❡ ✶✳✶✵✳✶ ✐♥ s❡❝t✐♦♥ ✶✳✶✵✱ ❈❤❛♣t❡r ✶✳

❊①❝❡r❝✐s❡ ✽✳✶✳✻ ❚❤❡ s❤♦rt❡st ❞✐❛❣♦♥❛❧ ♦❢ ❛ ♣❛r❛❧❧❡❧♦❣r❛♠ ❤❛s ❧❡♥❣t❤ ✽✱ t❤❡❛♥❣❧❡ ♦❢ t❤❡ ❞✐❛❣♦♥❛❧s ✐s ✹✺ ❞❡❣r❡❡✱ ❛♥❞ ✐ts ❛r❡❛ ✐s ✹✵✳ ❈❛❧❝✉❧❛t❡ t❤❡ ♣❡r✐♠❡t❡r♦❢ t❤❡ ♣❛r❛❧❧❡❧♦❣r❛♠✳

❙♦❧✉t✐♦♥✳ ❚❤❡ ❛r❡❛ ♠✉st ❜❡ t❤❡ s✉♠ ♦❢ t❤❡ ❛r❡❛s ♦❢ tr✐❛♥❣❧❡s ❆❋❇✱ ❇❋❈✱ ❈❋❉❛♥❞ ❉❋❆✳ ❚❤❡② ❛r❡ ♣❛✐r✇✐s❡ ❝♦♥❣r✉❡♥t ❛♥❞ ✇❡ ❛❧s♦ ❦♥♦✇ t❤❛t t❤❡ ❞✐❛❣♦♥❛❧s♦❢ ❛ ♣❛r❛❧❧❡❧♦❣r❛♠ ❜✐s❡❝t ❡❛❝❤ ♦t❤❡r✳ ■❢ ①❂❆❋❂❋❈ t❤❡♥

40 = 2x · 4 · sin 45

2+ 2

4 · x · sin(180− 45)

2.

❙✐♥❝❡ s✐♥ ✹✺ ❂ s✐♥ ✭✶✽✵ ✲ ✹✺✮ ✐t ❢♦❧❧♦✇s t❤❛t ① ❂ 10/√2✳ ❯s✐♥❣ t❤❡ ❝♦s✐♥❡ r✉❧❡

✐♥ t❤❡ tr✐❛♥❣❧❡ ❇❋❈

BC2 = 42 + x2 − 2 · 4 · x · cos 45 = 16 + 50− 40 = 26.

■♥ ❛ s✐♠✐❧❛r ✇❛②

AB2 = 42 + x2 − 2 · 4 · x · cos(180− 45) = 42 + x2 + 2 · 4 · x · cos 45 = 106.

❚❤❡r❡❢♦r❡ ❇❈≈✺✳✵✾✱ ❆❇≈✶✵✳✷✾ ❛♥❞ t❤❡ ♣❡r✐♠❡t❡r ✐s P≈✸✵✳✼✻✳

❊①❝❡r❝✐s❡ ✽✳✶✳✼ ❚❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ♠✐❞ ✲ ❧✐♥❡ ♦❢ ❛ s②♠♠❡tr✐❝ tr❛♣❡③✐✉♠ ✐s✶✵✱ t❤❡ ❞✐❛❣♦♥❛❧s ❛r❡ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ ❡❛❝❤ ♦t❤❡r✳ ❲❤❛t ✐s t❤❡ ❛r❡❛ ♦❢ t❤❡tr❛♣❡③✐✉♠✳

❙♦❧✉t✐♦♥✳ ❇❡❝❛✉s❡ ♦❢ t❤❡ s②♠♠❡tr②

x = DE = CE ❛♥❞ y = AE = BE.

❚❤❡ ♣❛r❛❧❧❡❧ ❜❛s❡s ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❜② t❤❡ P②t❤❛❣♦r❡❛♥ t❤❡♦r❡♠✿

CD =√2x ❛♥❞ AB =

√2y.

✽✳✶✳ ❊❳❊❘❈■❙❊❙ ✶✶✾

❋✐❣✉r❡ ✽✳✺✿ ❊①❡r❝✐s❡ ✽✳✶✳✼

❋✐❣✉r❡ ✽✳✻✿ ❊①❡r❝✐s❡ ✽✳✶✳✽

❙✐♥❝❡ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ♠✐❞ ✲ ❧✐♥❡ ✐s ✶✵ ✇❡ ❤❛✈❡ t❤❛t

10 =

√2x+

√2y

2.

❋r♦♠ ❤❡r❡

x+ y =20√2

❛♥❞ t❤❡ ❛r❡❛ ✐s

A =x2

2+

y2

2+ 2

xy

2=

(x+ y)2

2= 100.

❊①❝❡r❝✐s❡ ✽✳✶✳✽ ❚❤❡ ❞✐❛❣♦♥❛❧s ♦❢ ❛ tr❛♣❡③✐✉♠ ❛r❡ ♣❡r♣❡♥❞✐❝✉❧❛r✳ ❚❤❡ ❧❡♥❣t❤s♦❢ t❤❡ ♣❛r❛❧❧❡❧ s✐❞❡s ❛r❡ ✶✼ ❛♥❞ ✸✹✱ ♦♥❡ ♦❢ t❤❡ ❧❡❣s ✐s

√964✳ ❍♦✇ ❧♦♥❣ ✐s t❤❡

s❡❝♦♥❞ ❧❡❣✱ ✇❤❛t ✐s t❤❡ ❛r❡❛✱ ❛♥❞ t❤❡ ❤❡✐❣❤t ♦❢ t❤❡ tr❛♣❡③✐✉♠✳

❙♦❧✉t✐♦♥✳ ❚❤❡ tr✐❛♥❣❧❡s ❆❊❇ ❛♥❞ ❈❊❉ ❛r❡ s✐♠✐❧❛r✳ ❚❤❡ r❛t✐♦ ♦❢ t❤❡ s✐♠✐❧❛r✐t②✐s ❥✉st ✷❂✸✹✴✶✼✳ ❚❤❡r❡❢♦r❡

AE = 2 · EC ❛♥❞ BE = 2 ·DE.

✶✷✵ ❈❍❆P❚❊❘ ✽✳ ❊❳❊❘❈■❙❊❙

❙✉♣♣♦s❡ t❤❛tBC =

√964.

❇② P②t❤❛❣♦r❡❛♥ t❤❡♦r❡♠ ✐♥ t❤❡ r✐❣❤t tr✐❛♥❣❧❡s ❇❊❈ ❛♥❞ ❈❊❉✿

BE2 + CE2 = 964 ⇒ 4 ·DE2 + CE2 = 964,

CE2 +DE2 = 172.

❚❤❡r❡❢♦r❡3 ·DE2 = 675 ⇒ DE = 15

❛♥❞ ❈❊❂✽✳ ❚❤✐s ♠❡❛♥s t❤❛t ❆❊❂✶✻ ❛♥❞ ❇❊❂✸✵✳ ❚❤❡ s❡❝♦♥❞ ❧❡❣ ✐s

AD =√AE2 +DE2 =

√481.

❚❤❡ ❛r❡❛ ✐s

A =AE · BE

2+

BE · CE

2+

CE ·DE

2+

DE · AE2

= 540.

❙✐♥❝❡

540 =34 + 17

2m

t❤❡ ❤❡✐❣❤t ♦❢ t❤❡ tr❛♣❡③✐✉♠ ✐s ♠ ❂ ✶✵✽✵✴✺✶ ❂ ✸✻✵✴✶✼✳

❊①❝❡r❝✐s❡ ✽✳✶✳✾ ❚❤❡ ♣❛r❛❧❧❡❧ ❜❛s❡s ♦❢ ❛ s②♠♠❡tr✐❝❛❧ tr❛♣❡③♦✐❞ ❛r❡ ✶✵ ❛♥❞✷✵✳ ❚❤❡ ❤❡✐❣❤t ✐s ✹✳

• ❈❛❧❝✉❧❛t❡ t❤❡ ❛r❡❛ ♦❢ t❤❡ tr❛♣❡③♦✐❞✳

• ❈❛❧❝✉❧❛t❡ t❤❡ ❛♥❣❧❡s ♦❢ t❤❡ tr❛♣❡③♦✐❞✳

❊①❝❡r❝✐s❡ ✽✳✶✳✶✵ ❚❤❡ ❧♦♥❣❡st ❜❛s❡ ♦❢ ❛ s②♠♠❡tr✐❝❛❧ tr❛♣❡③♦✐❞ ✐s ✷✵✱ t❤❡❧❡♥❣t❤ ♦❢ t❤❡ ❧❡❣s ✐s ✺✱ t❤❡ ❤❡✐❣❤t ✐s ✹✳

• ❈❛❧❝✉❧❛t❡ t❤❡ ❛r❡❛ ♦❢ t❤❡ tr❛♣❡③♦✐❞✳

• ❈❛❧❝✉❧❛t❡ t❤❡ ❛♥❣❧❡s ♦❢ t❤❡ tr❛♣❡③♦✐❞✳

❊①❝❡r❝✐s❡ ✽✳✶✳✶✶ ■♥ ❦✐t❡ ❆❇❈❉ ✇❡ ❦♥♦✇ t❤❛t ❆❇ ❂ ❇❈ ❂ ✷ ❛♥❞ ❈❉ ❂ ❉❆✳❆t ✈❡rt❡① ❆ t❤❡ ❛♥❣❧❡ ✐s ✶✷✵ ❞❡❣r❡❡✱ ❛♥❞ ❛t ❉ t❤❡ ❛♥❣❧❡ ✐s ✻✵ ❞❡❣r❡❡✳ ❈❛❧❝✉❧❛t❡t❤❡ ✉♥❦♥♦✇♥ ❛♥❣❧❡s✱ s✐❞❡s ❛♥❞ ❞✐❛❣♦♥❛❧s ♦❢ t❤❡ ❦✐t❡ ❛♥❞ ❢✉rt❤❡r♠♦r❡✱ t❤❡r❛❞✐✉s ♦❢ t❤❡ ✐♥s❝r✐❜❡❞ ❝✐r❝❧❡✳

✽✳✶✳ ❊❳❊❘❈■❙❊❙ ✶✷✶

❋✐❣✉r❡ ✽✳✼✿ ❊①❡r❝✐s❡ ✽✳✶✳✶✶

❙♦❧✉t✐♦♥✳ ❙✐♥❝❡ ✇❡ ❤❛✈❡ ❡q✉❛❧ ❛❞❥❛❝❡♥t s✐❞❡s ✐t ❢♦❧❧♦✇s t❤❛t ❜♦t❤ ❆❈❉ ❛♥❞❆❇❈ ❛r❡ ❡q✉✐❧❛t❡r❛❧ tr✐❛♥❣❧❡s✳ ❚❤❡r❡❢♦r❡ ✇❡ ❤❛✈❡ ❛ r❤♦♠❜✉s ✇✐t❤ s✐❞❡s ♦❢❧❡♥❣t❤ ✷✳ ❚❤❡ ❛♥❣❧❡s ❛r❡ ❛❧t❡r♥❛t❡❧② ✻✵ ❛♥❞ ✶✷✵ ❞❡❣r❡❡ ✐♥ ♠❡❛s✉r❡✳ ❙✐♥❝❡❆❇❈ ✐s ❛♥ ❡q✉✐❧❛t❡r❛❧ tr✐❛♥❣❧❡ t❤❡ ❞✐❛❣♦♥❛❧ ❆❈ ✐s ✷ t♦♦✳ ❚♦ ❝♦♠♣✉t❡ t❤❡❧❡♥❣t❤ ♦❢ t❤❡ ❧♦♥❣❡r ❞✐❛❣♦♥❛❧ ✇❡ ❝❛♥ ✉s❡ t❤❡ ❝♦s✐♥❡ r✉❧❡

BD2 = 22 + 22 − 2 · 2 · 2 · cos 120 = 12.

❚❤❡ r❛❞✐✉s ♦❢ t❤❡ ✐♥s❝r✐❜❡❞ ❝✐r❝❧❡ ✐s ❥✉st

r =A

s,

✇❤❡r❡ t❤❡ s❡♠✐♣❡r✐♠❡t❡r s ✐s ✹✳ ❚♦ ❝♦♠♣✉t❡ ❆ ✇❡ ✉s❡ t❤❡ tr✐❣♦♥♦♠❡tr✐❝❢♦r♠✉❧❛ ❢♦r t❤❡ ❛r❡❛

A = 2 · 2 · sin 60 = 2√3.

❚❤❡r❡❢♦r❡

r =

√3

2.

❊①❝❡r❝✐s❡ ✽✳✶✳✶✷ ❚❤❡ ♣❡r✐♠❡t❡r ♦❢ t❤❡ r❤♦♠❜✉s ✐s ✹✵✱ ✐ts ❛r❡❛ ✐s ✾✻✳ ❲❤❛t❛r❡ t❤❡ ❛♥❣❧❡s✱ s✐❞❡s✱ ❛♥❞ ❞✐❛❣♦♥❛❧s ♦❢ t❤❡ r❤♦♠❜✉s✳

❙♦❧✉t✐♦♥✳ ■❢ ❛ ❞❡♥♦t❡s t❤❡ ❝♦♠♠♦♥ ❧❡♥❣t❤ ♦❢ t❤❡ s✐❞❡s ♦❢ t❤❡ r❤♦♠❜✉s t❤❡♥✹✵ ❂ ✹❛✱ ✐✳❡✳ ❛ ❂ ✶✵✳ ❚♦ ❝♦♠♣✉t❡ t❤❡ ❛r❡❛ ✇❡ ❝❛♥ ✇r✐t❡ t❤❛t

96 = 102 sinα

✶✷✷ ❈❍❆P❚❊❘ ✽✳ ❊❳❊❘❈■❙❊❙

❛♥❞✱ ❝♦♥s❡q✉❡♥t❧② sinα = 0.96✳ ❚❤✐s ♠❡❛♥s t❤❛t t❤❡ ❛♥❣❧❡s ❛r❡ α1 ≈ 73.74❛♥❞ α2 = 180−α1 ≈ 106.26✳ ❯s✐♥❣ t❤❡ ❝♦s✐♥❡ r✉❧❡ s②st❡♠❛t✐❝❛❧❧② t❤❡ ❧❡♥❣t❤♦❢ t❤❡ ❞✐❛❣♦♥❛❧s ❛r❡

d1 =√

102 + 102 − 2 · 10 · 10 · cosα1

d2 =√

102 + 102 − 2 · 10 · 10 · cosα2 =√

102 + 102 + 2 · 10 · 10 · cosα1.

❙✐♥❝❡ α1 ✐s ❛♥ ❛❝✉t❡ ❛♥❣❧❡

cosα1 =√1− 0.962 = 0.28.

❚❤❡r❡❢♦r❡d1 =

√144 = 12 ❛♥❞ d2 =

√256 = 16.

❊①❝❡r❝✐s❡ ✽✳✶✳✶✸ ❚❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ t✇♦ ❞✐❛❣♦♥❛❧s ♦❢ ❛ r❤♦♠❜✉s ❛r❡ ❣✐✈❡♥✿✻ ❛♥❞ ✶✷✳

• ❈❛❧❝✉❧❛t❡ t❤❡ ❛r❡❛ ♦❢ t❤❡ r❤♦♠❜✉s✦

• ❈❛❧❝✉❧❛t❡ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ s✐❞❡s ♦❢ t❤❡ r❤♦♠❜✉s✦

• ❈❛❧❝✉❧❛t❡ t❤❡ ❛♥❣❧❡s ♦❢ t❤❡ r❤♦♠❜✉s✦

❊①❝❡r❝✐s❡ ✽✳✶✳✶✹ ❚❤❡ ❧♦♥❣❡r ❞✐❛❣♦♥❛❧ ♦❢ ❛ r❤♦♠❜✉s ✐s ❣✐✈❡♥✿ ✶✷✱ ❛♥❞ ♦♥❡♦❢ t❤❡ ❛♥❣❧❡ ♦❢ t❤❡ r❤♦♠❜✉s ✐s ✻✵ ❞❡❣r❡❡ ✐♥ ♠❡❛s✉r❡✳

• ❈❛❧❝✉❧❛t❡ t❤❡ ❛r❡❛ ♦❢ t❤❡ r❤♦♠❜✉s✳

• ❈❛❧❝✉❧❛t❡ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ s✐❞❡s ♦❢ t❤❡ r❤♦♠❜✉s✳

❊①❝❡r❝✐s❡ ✽✳✶✳✶✺ ❚❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ s✐❞❡ ♦❢ ❛ r❤♦♠❜✉s ✐s ❥✉st t❤❡ ❣❡♦♠❡tr✐❝♠❡❛♥ ♦❢ t❤❡ ❞✐❛❣♦♥❛❧s✳ ❲❤❛t ✐s t❤❡ r❛t✐♦ ♦❢ t❤❡ t✇♦ ❞✐❛❣♦♥❛❧s✳

❙♦❧✉t✐♦♥✳ ▲❡t ❡ ❛♥❞ ❢ ❜❡ t❤❡ ❧❡♥❣t❤s ♦❢ t❤❡ ❞✐❛❣♦♥❛❧s✳ ❚❤❡ ❞✐❛❣♦♥❛❧s ♦❢ ❛r❤♦♠❜✉s ❛r❡ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ ❡❛❝❤ ♦t❤❡r ❜❡❝❛✉s❡ ♦❢ t❤❡ ❣❡♦♠❡tr✐❝ ❝❤❛r❛❝t❡r✲✐③❛t✐♦♥ ♦❢ t❤❡ ♣❡r♣❡♥❞✐❝✉❧❛r ❜✐s❡❝t♦r✳ ❚❤❡r❡❢♦r❡

a2 =e2

4+

f 2

4.

❖♥ t❤❡ ♦t❤❡r ❤❛♥❞ a2 = ef ✳ ❚❤✐s ♠❡❛♥s t❤❛t

ef =e2

4+

f 2

4

✽✳✶✳ ❊❳❊❘❈■❙❊❙ ✶✷✸

❋✐❣✉r❡ ✽✳✽✿ ●❡♦♠❡tr✐❝ ♣r♦❜❛❜✐❧✐t②

❛♥❞

4 = x+1

x,

✇❤❡r❡ ① ❂ ❡ ✿ ❢✳ ❚❤❡r❡❢♦r❡

0 = x2 − 4x+ 1

❛♥❞

x12 =4±

√12

2= 2±

√3.

❊①❝❡r❝✐s❡ ✽✳✶✳✶✻ Pr♦✈❡ t❤❛t

2 +√3 =

1

2−√3.

❊①❝❡r❝✐s❡ ✽✳✶✳✶✼ ❚✇♦ ♣❡rs♦♥s ❛r❡ ❣♦✐♥❣ t♦ ♠❡❡t ✇✐t❤✐♥ ♦♥❡ ❤♦✉r✳ ❚❤❡②❛❣r❡❡ t❤❛t ❛♥② ♦❢ t❤❡♠ ✇✐❧❧ ✇❛✐t ❢♦r t❤❡ ♦t❤❡r ❛t ♠♦st ✷✵ ♠✐♥✉t❡s✳ ❲❤❛t ✐st❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡ ♠❡❡t✐♥❣✳

❙♦❧✉t✐♦♥✳ ❋✐rst ♦❢ ❛❧❧ ✇❡ s❤♦✉❧❞ ✜♥❞ ❛ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ♦❢ t❤❡ ♣r♦❜❧❡♠✳▲❡t ① ❛♥❞ ② ❜❡ t❤❡ ❛rr✐✈✐♥❣ t✐♠❡ ♦❢ ♣❡rs♦♥s ❆ ❛♥❞ ❇✱ r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡s❡ ❛r❡r❛♥❞♦♠❧② ❝❤♦s❡♥ ❢r♦♠ t❤❡ ✐♥t❡r✈❛❧ ❬✵✱✶❪✳ ■♥ ♦t❤❡r ✇♦r❞s ❛♥② ❡✈❡♥t ❝♦rr❡s♣♦♥❞t♦ ❛ ♣♦✐♥t P✭①✱②✮ ♦❢ t❤❡ sq✉❛r❡ ✇✐t❤ s✐❞❡s ♦❢ ✉♥✐t ❧❡♥❣t❤✳ ❆ ❛♥❞ ❇ ♠❡❡t ✐❢❛♥❞ ♦♥❧② ✐❢ t❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ ② ✲ ① ✐s ❧❡ss ♦r ❡q✉❛❧ t❤❛♥ ✵✳✸❤♦✉r❂✷✵ ♠✐♥✉t❡s✳ ❲❡ ❛r❡ ❣♦✐♥❣ t♦ ❝♦♠♣✉t❡ ✇❤❛t ✐s t❤❡ ❛r❡❛ ♦❢ t❤❡ s❡t ♦❢♣♦✐♥ts s❛t✐s❢②✐♥❣ t❤❡ ✐♥❡q✉❛❧✐t✐❡s

−0.3 ≤ y − x ≤ 0.3.

✶✷✹ ❈❍❆P❚❊❘ ✽✳ ❊❳❊❘❈■❙❊❙

❚❤❡s❡ ♣♦✐♥ts r❡♣r❡s❡♥t s✉❝❝❡ss❢✉❧ ♦✉t❝♦♠❡s✳ ❯s✐♥❣ t❤❡ ❛r❡❛ ✇❤✐❝❤ ✐s ♠✐ss✐♥❣✿

❚❤❡ ❛r❡❛ ♦❢ s✉❝❝❡ss❢✉❧❧ ♦✉t❝♦♠❡s = 1− (1− 0.3)2

2− (1− 0.3)2

2= 0.51.

❚❤❡r❡❢♦r❡ t❤❡ ♣r♦❜❛❜✐❧✐t② ✐s

P =t❤❡ ❛r❡❛ ♦❢ s✉❝❝❡ss❢✉❧ ♦✉t❝♦♠❡st❤❡ ❛r❡❛ ♦❢ ❛❧❧ t❤❡ ♦✉t❝♦♠❡s

= 0.51.

❈❤❛♣t❡r ✾

P♦❧②❣♦♥s

✾✳✶ P♦❧②❣♦♥s

■♥ ❣❡♥❡r❛❧ ♣♦❧②❣♦♥s ❛r❡ ♣❧❛♥❡ ✜❣✉r❡s ❜♦✉♥❞❡❞ ❜② ❛ ✜♥✐t❡ ❝❤❛✐♥ ♦❢ str❛✐❣❤t❧✐♥❡ s❡❣♠❡♥ts✳ ❙✐♥❝❡ t❤❡② ❛r❡ t②♣✐❝❛❧❧② ✐♥✈❡st✐❣❛t❡❞ ❜② ✉s✐♥❣ ❛ tr✐❛♥❣❧❡ ❞❡✲❝♦♠♣♦s✐t✐♦♥ ✇❡ ❛❣r❡❡ t❤❛t ❛♥② tr✐❛♥❣❧❡ ✐s ❛ ♣♦❧②❣♦♥✳

❉❡✜♥✐t✐♦♥ ❆ s✐♠♣❧❡ ❝❧♦s❡❞ ♣♦❧②❣♦♥ ✐s ❛ ✜♥✐t❡ ✉♥✐♦♥ ♦❢ ❧✐♥❡ s❡❣♠❡♥ts

A1A2, A2A3, . . . , AnAn+1,

✇❤❡r❡ A1✱ ✳✳✳✱ An ❛r❡ ❞✐st✐♥❝t ♣♦✐♥ts ✐♥ t❤❡ ♣❧❛♥❡✱ A1 = An+1 ❛♥❞ t❤❡ ❧✐♥❡s❡❣♠❡♥ts ❤❛✈❡ ♥♦ ♦t❤❡r ♣♦✐♥ts ✐♥ ❝♦♠♠♦♥ ❡①❝❡♣t t❤❡✐r ❡♥❞♣♦✐♥ts✱ ❡❛❝❤ ♦❢✇❤✐❝❤ ❧✐❡s ♦♥ t✇♦ s❡❣♠❡♥ts✳

❚❤❡ ❜♦✉♥❞❛r② ♦❢ s✉❝❤ ❛ s❤❛♣❡ ✐s ❛ ❝❤❛✐♥ ♦❢ str❛✐❣❤t ❧✐♥❡ s❡❣♠❡♥ts✳ ❚❤❡♣♦s✐t✐♦♥s ✇❤❡r❡ t❤❡ ❝❤❛✐♥ ✐s ❜r♦❦❡♥ ❛t ❛r❡ ❝❛❧❧❡❞ ✈❡rt✐❝❡s✳ ❚❤❡ str❛✐❣❤t ❧✐♥❡s❡❣♠❡♥t ❜❡t✇❡❡♥ t✇♦ ❛❞❥❛❝❡♥t ✈❡rt✐❝❡s ✐s ❝❛❧❧❡❞ ❛ s✐❞❡✴❡❞❣❡✶ ♦❢ t❤❡ ♣♦❧②❣♦♥✳❚❤❡ ♣♦❧②❣♦♥ ✐s ❝❛❧❧❡❞ ❝♦♥✈❡① ✐❢ t❤❡r❡ ❛r❡ ♥♦ ❝♦♥❝❛✈❡ ✐♥t❡r✐♦r ❛♥❣❧❡s✱ ✐✳❡✳ ❛❧❧t❤❡ ✐♥t❡r✐♦r ❛♥❣❧❡s ❛r❡ ♦❢ ♠❡❛s✉r❡ ❧❡ss t❤❛♥ ✶✽✵ ❞❡❣r❡❡✳

❚❤❡♦r❡♠ ✾✳✶✳✶ ❚❤❡ s✉♠ ♦❢ ✐♥t❡r✐♦r ❛♥❣❧❡s ♦❢ ❛ ♣♦❧②❣♦♥ ❤❛✈✐♥❣ ♥ s✐❞❡s ✐s

(n− 2)π.

Pr♦♦❢ ■♥ ❝❛s❡ ♦❢ ❝♦♥✈❡① ♣♦❧②❣♦♥s t❤❡ r❡s✉❧t ❢♦❧❧♦✇s ❡❛s✐❧② ❢r♦♠ t❤❡ tr✐❛♥❣❧❡❞❡❝♦♠♣♦s✐t✐♦♥✳ ❖t❤❡r✇✐s❡ t❤❡ st❛t❡♠❡♥t ❝❛♥ ❜❡ ♣r♦✈❡❞ ❜② ✐♥❞✉❝t✐♦♥ ♦♥ t❤❡♥✉♠❜❡r ♦❢ ❝♦♥❝❛✈❡ ✐♥t❡r✐♦r ❛♥❣❧❡s✳

✶❚❤❡ ♥❛♠❡ ❡❞❣❡ ✐s t②♣✐❝❛❧❧② ✉s❡❞ ✐♥ ❣r❛♣❤ t❤❡♦r②✳

✶✷✺

✶✷✻ ❈❍❆P❚❊❘ ✾✳ P❖▲❨●❖◆❙

❋✐❣✉r❡ ✾✳✶✿ ❆ ♣♦❧②❣♦♥

❚❤❡ s✉♠ ♦❢ t❤❡ ❞✐❛❣♦♥❛❧s✳ ❙✉♣♣♦s❡ t❤❛t t❤❡ ♣♦❧②❣♦♥ ❤❛s ♥ ❃ ✸ ✈❡rt✐❝❡s✳■❢ ❆ ✐s ♦♥❡ ♦❢ t❤❡♠ t❤❡♥ ✇❡ ❤❛✈❡ ♥ ✲ ✶ ✈❡rt✐❝❡s ❧❡❢t t♦ ❥♦✐♥ ✇✐t❤ ❆✳ ❚❤❡s❡ ❣✐✈❡t✇♦ s✐❞❡s ✭❛❞❥❛❝❡♥t ✈❡rt✐❝❡s✮ ❛♥❞ ♥ ✲ ✸ ❞✐❛❣♦♥❛❧s✳ ❚❤❡r❡❢♦r❡ t❤❡ s✉♠ ♦❢ t❤❡❞✐✛❡r❡♥t ❞✐❛❣♦♥❛❧s ♦❢ t❤❡ ♣♦❧②❣♦♥ ❤❛✈✐♥❣ ♥ ✈❡rt✐❝❡s ✐s

n(n− 3)

2

❜❡❝❛✉s❡ ❡❛❝❤ ❞✐❛❣♦♥❛❧ ❜❡❧♦♥❣s t♦ ❡①❛❝t❧② t✇♦ ✈❡rt✐❝❡s✳❚❤❡ ❛r❡❛ ♦❢ ❛ ♣♦❧②❣♦♥ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❛s t❤❡ s✉♠ ♦❢ t❤❡ ❛r❡❛s ♦❢ tr✐❛♥❣❧❡s❝♦♥st✐t✉t✐♥❣ t❤❡ ♣♦❧②❣♦♥✳

❖♥❡ ♦❢ t❤❡ ♠♦st ✐♠♣♦rt❛♥t s♣❡❝✐❛❧ ❝❧❛ss❡s ♦❢ ♣♦❧②❣♦♥s ✐s ❢♦r♠❡❞ ❜② r❡❣✉❧❛r♣♦❧②❣♦♥s✳ ❚❤❡② ❛r❡ ❛✉t♦♠❛t✐❝❛❧❧② ✐♥s❝r✐❜❡❞ ✐♥ ❛ ❝✐r❝❧❡ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✇❛②✳▲❡t ❛ ❝✐r❝❧❡ ❜❡ ❣✐✈❡♥ ❛♥❞ ❞✐✈✐❞❡ t❤❡ ♣❡r✐♠❡t❡r ✐♥t♦ ♥ ❡q✉❛❧ ♣❛rts ❜② t❤❡ ♣♦✐♥ts

P1, P2, . . . , Pn,

✇❤❡r❡ ♥ ✐s ❣❡❛t❡r ♦r ❡q✉❛❧ t❤❛♥ ✸✳ ❊❛❝❤ ♦❢ t❤❡ ❝❤♦r❞s

P1P2, P2P3, . . . , PnP1

❜❡❧♦♥❣ t♦ t❤❡ ❝❡♥tr❛❧ ❛♥❣❧❡ ✸✻✵✴♥ ❞❡❣r❡❡ ✐♥ ♠❡❛s✉r❡✳ ❚❤❡② ❛r❡ t❤❡ s✐❞❡s♦❢ t❤❡ r❡❣✉❧❛r ♥✕❣♦♥ ✐♥s❝r✐❜❡❞ ✐♥ t❤❡ ❣✐✈❡♥ ❝✐r❝❧❡✳ ❚❤❡ s✐③❡ ❞❡♣❡♥❞s ♦♥ t❤❡r❛❞✐✉s ♦❢ t❤❡ ❝✐r❝❧❡✳

❈❤❛♣t❡r ✶✵

❈✐r❝❧❡s

❉❡✜♥✐t✐♦♥ ▲❡t ❛ ♣♦✐♥t ❖ ✐♥ t❤❡ ♣❧❛♥❡ ❜❡ ❣✐✈❡♥✳ ■❢ r ✐s ❛ ♣♦s✐t✐✈❡ r❡❛❧ ♥✉♠❜❡rt❤❡♥ t❤❡ s❡t ♦❢ ♣♦✐♥ts ❤❛✈✐♥❣ ❞✐st❛♥❝❡ r ❢r♦♠ ❖ ✐s ❝❛❧❧❡❞ ❛ ❝✐r❝❧❡✳ ❚❤❡ ♣♦✐♥t ❖✐s t❤❡ ❝❡♥t❡r ❛♥❞ r ✐s t❤❡ r❛❞✐✉s ♦❢ t❤❡ ❝✐r❝❧❡✳ ❆ ❞✐s❦ ♠❡❛♥s t❤❡ s❡t ♦❢ ♣♦✐♥ts❤❛✈✐♥❣ ❞✐st❛♥❝❡ ❛t ♠♦st r ❢r♦♠ t❤❡ ❣✐✈❡♥ ♣♦✐♥t ❖✳ ❚❤❡ ❝✐r❝❧❡ ✐s t❤❡ ❜♦✉♥❞❛r②♦❢ t❤❡ ❞✐s❦ ✇✐t❤ t❤❡ s❛♠❡ ❝❡♥t❡r ❛♥❞ r❛❞✐✉s✳

❚❤❡ ♠♦st ✐♠♣♦rt❛♥t ♣r♦❜❧❡♠s r❡❧❛t❡❞ t♦ ❛ ❝✐r❝❧❡ ✐s t❤❡ ♣r♦❜❧❡♠ ♦❢ t❛♥❣❡♥t❧✐♥❡s ❛♥❞ t❤❡ ♣r♦❜❧❡♠ ♦❢ ❛r❡❛✳

✶✵✳✶ ❚❛♥❣❡♥t ❧✐♥❡s

▲❡t ❛ ❧✐♥❡ ❧ ❜❡ ❣✐✈❡♥✳ ❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❝✐r❝❧❡ s✉❣❣❡sts ✉s t♦ ❝❧❛ss✐❢②t❤❡ ♣♦✐♥ts ♦❢ t❤❡ ❧✐♥❡ ❜② t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ ❝❡♥t❡r ♦❢ t❤❡ ❝✐r❝❧❡✳ ❆t ✜rsts✉♣♣♦s❡ t❤❛t ❧ ❞♦❡s ♥♦t ♣❛ss t❤❡ ❝❡♥t❡r ❖ ❛♥❞ ❝♦♥s✐❞❡r t❤❡ ❧✐♥❡ ❡ ♣❛ss✐♥❣t❤r♦✉❣❤ ❖ s✉❝❤ t❤❛t ❡ ✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ ❧✳ ❯s✐♥❣ P②t❤❛❣♦r❡❛♥ t❤❡♦r❡♠ ✐t❝❛♥ ❜❡ ❡❛s✐❧② s❡❡♥ t❤❛t t❤❡ ❢♦♦t ❋ ❤❛s t❤❡ s♠❛❧❧❡st ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ ❝❡♥t❡r❛♠♦♥❣ t❤❡ ♣♦✐♥ts ♦❢ ❧✳ ❚❤❡r❡❢♦r❡ ✐❢

• ❖❋❂r t❤❡♥ t❤❡ ❧✐♥❡ ❤❛s ❡①❛❝t❧② ♦♥❡ ❝♦♠♠♦♥ ♣♦✐♥t ✇✐t❤ t❤❡ ❝✐r❝❧❡ ❛♥❞❛❧❧ t❤❡ ♦t❤❡r ♣♦✐♥ts ❛r❡ ❡①t❡r♥❛❧✳ ■♥ t❤✐s ❝❛s❡ ✇❡ s❛② t❤❛t t❤❡ ❧✐♥❡ ✐st❛♥❣❡♥t t♦ t❤❡ ❝✐r❝❧❡ ❛t t❤❡ ♣♦✐♥t ❋ ♦❢ t❛♥❣❡♥❝②✳

• ❖❋ ❁ r t❤❡♥ t❤❡ ❧✐♥❡ ✐♥t❡rs❡❝t t❤❡ ❝✐r❝❧❡ ❛t ❡①❛❝t❧② t✇♦ ♣♦✐♥ts✳ ■♥ t❤✐s❝❛s❡ ✇❡ s♣❡❛❦ ❛❜♦✉t ❛ s❡❝❛♥t ❧✐♥❡✳

• ❖❋ ❃ r t❤❡♥ ❧ ❤❛s ♥♦ ♣♦✐♥ts ✐♥ ❝♦♠♠♦♥ ✇✐t❤ t❤❡ ❝✐r❝❧❡✳

❚❤❡ ❞✐s❝✉ss✐♦♥ ♦❢ t❤❡ ❧✐♥❡s ♣❛ss✐♥❣ t❤r♦✉❣❤ t❤❡ ❝❡♥t❡r ✐s ♦❜✈✐♦✉s✳

✶✷✼

✶✷✽ ❈❍❆P❚❊❘ ✶✵✳ ❈■❘❈▲❊❙

❉❡✜♥✐t✐♦♥ ❚❤❡ ❧✐♥❡ ❧ ✐s t❛♥❣❡♥t t♦ ❛ ❝✐r❝❧❡ ✐❢ t❤❡② ❤❛✈❡ ❡①❛❝t❧② ♦♥❡ ❝♦♠♠♦♥♣♦✐♥t ❛♥❞ ❛❧❧ t❤❡ ♦t❤❡r ♣♦✐♥ts ♦♥ t❤❡ ❧✐♥❡ ❛r❡ ❡①t❡r♥❛❧✳

❘❡♠❛r❦ ❆❧t❤♦✉❣❤ t❤❡ ❝♦♥❞✐t✐♦♥ ❛❧❧ t❤❡ ♦t❤❡r ♣♦✐♥ts ♦♥ t❤❡ ❧✐♥❡ ❛r❡ ❡①t❡r♥❛❧✐s r❡❞✉♥❞❛♥t ✐♥ ❝❛s❡ ♦❢ t❛♥❣❡♥t ❧✐♥❡s t♦ ❛ ❝✐r❝❧❡ ❜✉t ♥♦t ✐♥ ❣❡♥❡r❛❧ ❛s t❤❡ ❝❛s❡♦❢ ❝♦♥✐❝ s❡❝t✐♦♥s ✭❡❧❧✐♣s❡✱ ❤②♣❡r❜♦❧❛✱ ♣❛r❛❜♦❧❛✮ s❤♦✇s✳ ❚♦ ❝♦♥str✉❝t t❛♥❣❡♥t❧✐♥❡s ✐♥ ❣❡♥❡r❛❧ ♦♥❡ ♥❡❡❞ t❛❦✐♥❣ t❤❡ ❧✐♠✐t ❛❣❛✐♥✳ ❚❤❡ t❛♥❣❡♥t ❧✐♥❡ ✐s t❤❡ ❧✐♠✐t♣♦s✐t✐♦♥ ♦❢ ❝❤♦r❞s ♣❛ss✐♥❣ t❤r♦✉❣❤ ❛ ❣✐✈❡♥ ♣♦✐♥t ♦❢ t❤❡ ❝✉r✈❡✳

❊①❝❡r❝✐s❡ ✶✵✳✶✳✶ ❋✐♥❞ t❤❡ t❛♥❣❡♥t ❧✐♥❡s t♦ t❤❡ ♣❛r❛❜♦❧❛ ❣✐✈❡♥ ❜② t❤❡ ❣r❛♣❤♦❢ t❤❡ ❢✉♥❝t✐♦♥

f(x) = x2.

❙♦❧✉t✐♦♥✳ ▲❡t x = 1 ❜❡ ✜①❡❞ ❛♥❞ ❝♦♥s✐❞❡r t❤❡ ❝❤♦r❞ ♣❛ss✐♥❣ t❤r♦✉❣❤ t❤❡♣♦✐♥ts

(1, 1) ❛♥❞ (x, x2).

❚❤❡ s❧♦♣❡

m(x) =x2 − 1

x− 1

✐s ♦❜✈✐♦✉s❧② ❞❡♣❡♥❞ ♦♥ ①✳ ❲❤❛t ❤❛♣♣❡♥s ✐❢ ① t❡♥❞s t♦ ✶✳ ❙✐♥❝❡ t❤❡ ❞✐✈✐s✐♦♥❜② ③❡r♦ ✐s ✐♠♣♦ss✐❜❧❡ ✇❡ ❤❛✈❡ t♦ ❡❧✐♠✐♥❛t❡ t❤❡ t❡r♠ ① ✲ ✶✳ ❙✐♥❝❡

m(x) =x2 − 1

x− 1=

(x− 1)(x+ 1)

x− 1= x+ 1

t❤❡ s❧♦♣❡ ❛t t❤❡ ❧✐♠✐t ♣♦s✐t✐♦♥ ♠✉st ❜❡ ✷✳ ❚❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ t❛♥❣❡♥t ❧✐♥❡❛t ①❂✶ ✐s

y − f(1) = 2 · (x− 1) ⇒ y = 2x− 1.

❚❤❡♦r❡♠ ✶✵✳✶✳✷ ■❢ ❛ ❧✐♥❡ ✐s t❛♥❣❡♥t t♦ ❛ ❝✐r❝❧❡ t❤❡♥ ✐t ✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦t❤❡ r❛❞✐✉s ❞r❛✇♥ t♦ t❤❡ ♣♦✐♥t ♦❢ t❛♥❣❡♥❝②✳

❚❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ t❛♥❣❡♥t ❧✐♥❡ t♦ ❛ ❝✐r❝❧❡ ❢r♦♠ ❛ ❣✐✈❡♥ ❡①t❡r♥❛❧

♣♦✐♥t ✐s ❜❛s❡❞ ♦♥ ❚❤❛❧❡s✬ t❤❡♦r❡♠✳ ❙✉♣♣♦s❡ t❤❛t P ✐s ❛♥ ❡①t❡r♥❛❧ ♣♦✐♥t ❛♥❞❋ ✐s t❤❡ ♣♦✐♥t ♦❢ t❛♥❣❡♥❝② t♦ ❛ ❝✐r❝❧❡ ✇✐t❤ ❝❡♥t❡r ❖✳ ❙✐♥❝❡ P❋❖ ✐s ❛ r✐❣❤ttr✐❛♥❣❧❡ t❤❡ ♣♦✐♥t ❋ ♠✉st ❜❡ ♦♥ t❤❡ ♣❡r✐♠❡t❡r ♦❢ t❤❡ ❝✐r❝❧❡ ❞r❛✇♥ ❢r♦♠ t❤❡♠✐❞♣♦✐♥t ♦❢ t❤❡ s❡❣♠❡♥t ❖P ✇✐t❤ r❛❞✐✉s ❖P✴✷✳

❚❤❡♦r❡♠ ✶✵✳✶✳✸ ▲❡t ❛ ❝✐r❝❧❡ ✇✐t❤ ❝❡♥t❡r ❖ ❜❡ ❣✐✈❡♥ ❛♥❞ s✉♣♣♦s❡ t❤❛t P ✐s❛♥ ❡①t❡r♥❛❧ ♣♦✐♥t✳ ❚❤❡ t❛♥❣❡♥t ❧✐♥❡s ❢r♦♠ P t♦ t❤❡ ❝✐r❝❧❡ ❝❛♥ ❜❡ ❝♦♥str✉❝t❡❞❛s ❢♦❧❧♦✇s✿

✶✵✳✶✳ ❚❆◆●❊◆❚ ▲■◆❊❙ ✶✷✾

❋✐❣✉r❡ ✶✵✳✶✿ ❚❛♥❣❡♥t s❡❣♠❡♥ts ❢r♦♠ ❛♥ ❡①t❡r♥❛❧ ♣♦✐♥t

• ❞r❛✇ ❛ ❝✐r❝❧❡ ✇✐t❤ r❛❞✐✉s r ❂ ❖P✴✷ ❛r♦✉♥❞ t❤❡ ♠✐❞♣♦✐♥t ♦❢ ❖P✱

• t❤❡ ❝✐r❝❧❡ ❝♦♥str✉❝t❡❞ ✐♥ t❤❡ ✜rst st❡♣ ♠❡❡ts t❤❡ ❣✐✈❡♥ ❝✐r❝❧❡ ❛t t✇♦♣♦✐♥ts ❋ ❛♥❞ ●✱

• ❋P ❛♥❞ ●P ❛r❡ t❛♥❣❡♥t s❡❣♠❡♥ts t♦ t❤❡ ❣✐✈❡♥ ❝✐r❝❧❡✳

❚♦ ❝♦♠♣✉t❡ t❤❡ ❝♦♠♠♦♥ ❧❡♥❣t❤ ♦❢ t❤❡ t❛♥❣❡♥t s❡❣♠❡♥ts P❋ ❛♥❞ P● ✇❡ ❝❛♥✉s❡ P②t❤❛❣♦r❡❛♥ t❤❡♦r❡♠✿

PF 2 + r2 = OP 2.

❈♦r♦❧❧❛r② ✶✵✳✶✳✹ ❚❤❡ t❛♥❣❡♥t s❡❣♠❡♥ts ♣❛ss✐♥❣ t❤r♦✉❣❤ ❛ ❣✐✈❡♥ ❡①t❡r♥❛❧♣♦✐♥t ❛r❡ ♦❢ t❤❡ s❛♠❡ ❧❡♥❣t❤✳

❋♦r t✇♦ ❝✐r❝❧❡s t❤❡r❡ ❛r❡ ❣❡♥❡r❛❧❧② ❢♦✉r ❞✐st✐♥❝t s❡❣♠❡♥ts t❤❛t ❛r❡ t❛♥❣❡♥tt♦ ❜♦t❤ ♦❢ t❤❡♠✳ ■❢ t❤❡ ❝❡♥t❡rs ❛r❡ s❡♣❛r❛t❡❞ t❤❡♥ ✇❡ s♣❡❛❦ ❛❜♦✉t ✐♥t❡r♥❛❧❜✐t❛♥❣❡♥t s❡❣♠❡♥ts✳ ❖t❤❡r✇✐s❡ ✇❡ ❤❛✈❡ ❡①t❡r♥❛❧ ❜✐t❛♥❣❡♥t s❡❣♠❡♥ts✳ ■❢ t❤❡❝✐r❝❧❡s

• ❛r❡ ♦✉ts✐❞❡ ❡❛❝❤ ♦t❤❡r t❤❡♥ ✇❡ ❤❛✈❡ t✇♦ ❡①t❡r♥❛❧ ❛♥❞ t✇♦ ✐♥t❡r♥❛❧❜✐t❛♥❣❡♥t s❡❣♠❡♥ts s②♠♠❡tr✐❝❛❧❧② ❛❜♦✉t t❤❡ ❧✐♥❡ ♦❢ t❤❡ ❝❡♥t❡rs✳

• ❛r❡ t❛♥❣❡♥t t♦ ❡❛❝❤ ♦t❤❡r ❢r♦♠ ♦✉ts✐❞❡ t❤❡♥ ✇❡ ❤❛✈❡ ❛ ❝♦♠♠♦♥ ✭✐♥✲t❡r♥❛❧✮ t❛♥❣❡♥t ❧✐♥❡ ❛t t❤❡ ❝♦♥t❛❝t ♣♦✐♥t ❛♥❞ t✇♦ ❡①t❡r♥❛❧ ❜✐t❛♥❣❡♥ts❡❣♠❡♥ts s②♠♠❡tr✐❝❛❧❧② ❛❜♦✉t t❤❡ ❧✐♥❡ ♦❢ t❤❡ ❝❡♥t❡rs✳

• ✐♥t❡rs❡❝t ❡❛❝❤ ♦t❤❡r t❤❡♥ ✇❡ ❤❛✈❡ ♥♦ ✐♥♥❡r ❜✐t❛♥❣❡♥t s❡❣♠❡♥ts ♦r ❧✐♥❡s✳

• ❛r❡ t❛♥❣❡♥t t♦ ❡❛❝❤ ♦t❤❡r ❢r♦♠ ✐♥s✐❞❡ t❤❡♥ ✇❡ ❤❛✈❡ ♦♥❧② ❛ ❝♦♠♠♦♥✭❡①t❡r♥❛❧✮ t❛♥❣❡♥t ❧✐♥❡ ❛t t❤❡ ❝♦♥t❛❝t ♣♦✐♥t✳

❊①❝❡r❝✐s❡ ✶✵✳✶✳✺ ❍♦✇ t♦ ❝♦♥str✉❝t ❝♦♠♠♦♥ ❜✐t❛♥❣❡♥t s❡❣♠❡♥ts t♦ t✇♦ ❝✐r✲❝❧❡s❄

❙♦❧✉t✐♦♥✳ ❋♦r t❤❡ ❣❡♥❡r✐❝ ❝❛s❡s s❡❡ ✜❣✉r❡s ✶✵✳✷ ❛♥❞ ✶✵✳✸✳

✶✸✵ ❈❍❆P❚❊❘ ✶✵✳ ❈■❘❈▲❊❙

❋✐❣✉r❡ ✶✵✳✷✿ ❊①t❡r♥❛❧ ❜✐t❛♥❣❡♥t s❡❣♠❡♥ts

❋✐❣✉r❡ ✶✵✳✸✿ ■♥t❡r♥❛❧ ❜✐t❛♥❣❡♥t s❡❣♠❡♥ts

✶✵✳✷✳ ❚❆◆●❊◆❚■❆▲ ❆◆❉ ❈❨❈▲■❈ ◗❯❆❉❘■▲❆❚❊❘❆▲❙ ✶✸✶

✶✵✳✷ ❚❛♥❣❡♥t✐❛❧ ❛♥❞ ❝②❝❧✐❝ q✉❛❞r✐❧❛t❡r❛❧s

❘❡❣✉❧❛r ❣❡♦♠❡tr✐❝ ♦❜❥❡❝ts ❝❛♥ ❜❡ ❛❧✇❛②s ✐♠❛❣❡❞ t♦❣❡t❤❡r ✇✐t❤ t❤❡✐r ✐♥s❝r✐❜❡❞♦r ❝✐r❝✉♠s❝r✐❜❡❞ ❝✐r❝❧❡s✳ ❆♥♦t❤❡r t②♣❡ ♦❢ ♦❜❥❡❝ts ✐♥s❝r✐❜❡❞ ✐♥ ❛ ❝✐r❝❧❡ ❛r❡ t❤❡s♦ ✲ ❝❛❧❧❡❞ ❝②❝❧✐❝ q✉❛❞r✐❧❛t❡r❛❧s✳ ❚❤✐s ♠❡❛♥s t❤❛t t❤❡ ✈❡rt✐❝❡s ❛r❡ ❧②✐♥❣ ♦♥t❤❡ s❛♠❡ ❝✐r❝❧❡✳

❚❤❡♦r❡♠ ✶✵✳✷✳✶ ❚❤❡ q✉❛❞r✐❧❛t❡r❛❧ ❆❇❈❉ ✐s ❛ ❝②❝❧✐❝ q✉❛❞r✐❧❛t❡r❛❧ ✐❢ ❛♥❞♦♥❧② ✐❢ t❤❡ s✉♠s ♦❢ t❤❡ ♦♣♣♦s✐t❡ ❛♥❣❧❡s ❛r❡ ❡q✉❛❧✳

Pr♦♦❢ ❚❤❡ ♦♣♣♦s✐t❡ ❛♥❣❧❡s ♦❢ ❛ ❝②❝❧✐❝ q✉❛❞r✐❧❛t❡r❛❧ ❛r❡ ❧②✐♥❣ ♦♥ ❝♦♠♣❧❡♠❡♥t❛r❝s ✇❤✐❝❤ ♠❡❛♥s t❤❛t t❤❡ s✉♠ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❝❡♥tr❛❧ ❛♥❣❧❡s ✐s ✸✻✵❞❡❣r❡❡ ✐♥ ♠❡❛s✉r❡✳ ❚❤❡r❡❢♦r❡ t❤❡ s✉♠ ♦❢ t❤❡ ♦♣♣♦s✐t❡ ❛♥❣❧❡s ✐♥ ❛ ❝②❝❧✐❝q✉❛❞r✐❧❛t❡r❛❧ ♠✉st ❜❡ ✶✽✵ ❞❡❣r❡❡✳ ❈♦♥✈❡rs❡❧②✱ s✉♣♣♦s❡ t❤❛t ❢♦r ❡①❛♠♣❧❡

6 A = 180− 6 C

❛♥❞✱ ❝♦♥s❡q✉❡♥t❧②✱

sin 6 A = sin(180− 6 C) = sin 6 C.

❖♥ t❤❡ ♦t❤❡r ❤❛♥❞ t❤❡ tr✐❛♥❣❧❡s ❉❆❇ ❛♥❞ ❇❈❉ ❤❛✈❡ ❛ ❝♦♠♠♦♥ s✐❞❡ ❇❉✳❯s✐♥❣ t❤❡ ❡①t❡♥❞❡❞ s✐♥❡ r✉❧❡

a

sinα=

b

sin β=

c

sin γ= 2R

✇❡ ❤❛✈❡ t❤❛t t❤❡ r❛❞✐✉s ♦❢ t❤❡ ❝✐r❝✉♠s❝r✐❜❡❞ ❝✐r❝❧❡s ♦❢ t❤❡ tr✐❛♥❣❧❡s ❉❆❇❛♥❞ ❇❈❉ ♠✉st ❜❡ t❤❡ s❛♠❡✳ ❚❤❡ ❝✐r❝✉♠s❝r✐❜❡❞ ❝✐r❝❧❡s ♣❛ss s✐♠✉❧t❛♥❡♦✉s❧②t❤r♦✉❣❤ t❤❡ ♣♦✐♥ts ❇ ❛♥❞ ❉✳ ❚❤❡r❡❢♦r❡ t❤❡② ❛r❡ ❝♦✐♥❝✐❞❡ ♦r t❤❡ ✭❞✐✛❡r❡♥t✮❝❡♥t❡rs ❛r❡ s✐t✉❛t❡❞ s②♠♠❡tr✐❝❛❧❧② ❛❜♦✉t t❤❡ ❧✐♥❡ ♦❢ ❇❉ ❜❡❝❛✉s❡ ♦❢ t❤❡ ❝♦♠✲♠♦♥ r❛❞✐✉s✳ ❚❤✐s ✐s ✐♠♣♦ss✐❜❧❡ ❜❡❝❛✉s❡ t❤❡ ❛♥❣❧❡s 6 A ❛♥❞ 6 C ❝❛♥ ♥♦t ❜❡s✐♠✉❧t❛♥❡♦✉s❧② ❛❝✉t❡ ✭♦r ♦❜t✉s❡✮ ❛♥❣❧❡s✳

❉❡✜♥✐t✐♦♥ ❆ q✉❛❞r✐❧❛t❡r❛❧ ✐s ❝❛❧❧❡❞ t❛♥❣❡♥t✐❛❧ ✐❢ ✐t ❤❛s ❛♥ ✐♥s❝r✐❜❡❞ ❝✐r❝❧❡✇❤✐❝❤ t♦✉❝❤❡s ❛❧❧ t❤❡ s✐❞❡s ♦❢ t❤❡ q✉❛❞r✐❧❛t❡r❛❧✳

❚❤❡♦r❡♠ ✶✵✳✷✳✷ ❆ ❝♦♥✈❡① q✉❛❞r✐❧❛t❡r❛❧ ✐s t❛♥❣❡♥t✐❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ s✉♠♦❢ t❤❡ ♦♣♣♦s✐t❡ s✐❞❡s ❛r❡ ❡q✉❛❧✳

■❢ ✇❡ ❤❛✈❡ ❛ t❛♥❣❡♥t✐❛❧ q✉❛❞r✐❧❛t❡r❛❧ t❤❡♥ t❤❡ s✐❞❡s ❛r❡ ❝♦♥st✐t✉t❡❞ ❜②t❛♥❣❡♥t ❧✐♥❡ s❡❣♠❡♥ts t♦ t❤❡ ✐♥s❝r✐❜❡❞ ❝✐r❝❧❡✳ ■❢ ❊✱ ❋✱ ● ❛♥❞ ❍ ❞❡♥♦t❡ t❤❡

✶✸✷ ❈❍❆P❚❊❘ ✶✵✳ ❈■❘❈▲❊❙

❋✐❣✉r❡ ✶✵✳✹✿ ❈②❝❧✐❝ ❛♥❞ t❛♥❣❡♥t✐❛❧ q✉❛❞r✐❧❛t❡r❛❧s

t♦✉❝❤✐♥❣ ♣♦✐♥ts ♦♥ t❤❡ s✐❞❡s ❆❇✱ ❇❈✱ ❈❉ ❛♥❞ ❉❆ r❡s♣❡❝t✐✈❡❧②✱ t❤❡♥ ✇❡ ❤❛✈❡t❤❛t

AB + CD = (AE + EB) + (CG+GD) = (AH +BF ) + (FC +HD) =

= (AH +HD) + (BF + FC) = AD +BC

❜❡❝❛✉s❡ ❝♦r♦❧❧❛r② ✶✵✳✶✳✹ s❛②s t❤❛t t❤❡ t❛♥❣❡♥t ❧✐♥❡ s❡❣♠❡♥ts ❢r♦♠ ❛♥ ❡①t❡r♥❛❧♣♦✐♥t t♦ ❛ ❣✐✈❡♥ ❝✐r❝❧❡ ❛r❡ ♦❢ ❡q✉❛❧ ❧❡♥❣t❤✳ ❚❤❡r❡❢♦r❡ t❤❡ s✉♠ ♦❢ t❤❡ ❧❡♥❣t❤s♦❢ t❤❡ ♦♣♣♦s✐t❡ s✐❞❡s ❛r❡ ❡q✉❛❧✳ ❚❤❡ ❝♦♠♠♦♥ ✈❛❧✉❡ ✐s ♦❜✈✐♦✉s❧② t❤❡ ❤❛❧❢ ♦❢t❤❡ ♣❡r✐♠❡t❡r ♦❢ t❤❡ q✉❛❞r✐❧❛t❡r❛❧✳ ❚❤❡ ❝♦♥✈❡rs❡ st❛t❡♠❡♥t ❢❛✐❧s ✇✐t❤♦✉t t❤❡❝♦♥❞✐t✐♦♥ ♦❢ ❝♦♥✈❡①✐t② ❛s ❝♦♥❝❛✈❡ ❦✐t❡s s❤♦✇✳

✶✵✳✸ ❚❤❡ ❛r❡❛ ♦❢ ❝✐r❝❧❡s

❚♦ ❝♦♠♣✉t❡ t❤❡ ❛r❡❛ ♦❢ ❛ ❝✐r❝❧❡ ✇❡ ✉s❡ ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ❜❛s❡❞ ♦♥ ✐♥s❝r✐❜❡❞r❡❣✉❧❛r ♥ ✲ ❣♦♥s✳ ❋♦r t❤❡ s❛❦❡ ♦❢ s✐♠♣❧✐❝✐t② s✉♣♣♦s❡ t❤❛t t❤❡ ❝✐r❝❧❡ ❤❛sr❛❞✐✉s ♦♥❡✳ ❚❤❡ ✈❡rt✐❝❡s P1, P2, . . . , Pn ♦❢ ❛ r❡❣✉❧❛r ♥ ✲ ❣♦♥ ✐♥s❝r✐❜❡❞ ✐♥ ❛❝✐r❝❧❡ ❞✐✈✐❞❡s t❤❡ ♣❡r✐♠❡t❡r ✐♥t♦ ♥ ❡q✉❛❧ ♣❛rts✳ ❚❤❡r❡❢♦r❡ t❤❡ ❛r❡❛ ❝❛♥ ❜❡❝♦♠♣✉t❡❞ ❛s t❤❡ s✉♠ ♦❢ t❤❡ ❛r❡❛s ♦❢ t❤❡ ❝♦♥❣r✉❡♥t tr✐❛♥❣❧❡s

P1OP2, P2OP3, . . . , PnOP1,

✐✳❡✳

t❤❡ ❛r❡❛ ♦❢ ❛ r❡❣✉❧❛r ♥ ✲ ❣♦♥ ✐♥s❝r✐❜❡❞ ✐♥ t❤❡ ✉♥t✐ ❝✐r❝❧❡ = nsin 360

n

2.

❚♦ s✐♠♣❧✐❢② t❤❡ ♣r♦❝❡❞✉r❡ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❛r❡❛

Ak = 2ksin 360

2k

2= 2k−1 sin

360

2k

✶✵✳✸✳ ❚❍❊ ❆❘❊❆ ❖❋ ❈■❘❈▲❊❙ ✶✸✸

♦❢ 2k ✲ ❣♦♥s✳ ❲❡ ❛r❡ ❣♦✐♥❣ t♦ ❡①♣r❡ss t❤❡ ❛r❡❛

Ak+1 = 2k+1sin 360

2k+1

2= 2k sin

360

2k+1

✐♥ t❤❡ ✭❦✰✶✮t❤ st❡♣ ✐♥ t❡r♠s ♦❢ Ak✳ ❙✐♥❝❡

360

2k= 2

360

2k+1

✇❡ ❤❛✈❡ ❜② t❤❡ ❛❞❞✐t✐♦♥❛❧ r✉❧❡s t❤❛t

cos360

2k= cos

(

2360

2k+1

)

= cos2360

2k+1− sin2

360

2k+1= 1− 2 sin2

360

2k+1

❜❡❝❛✉s❡ ♦❢ t❤❡ tr✐❣♦♥♦♠❡tr✐❝ ✈❡rs✐♦♥ ♦❢ t❤❡ P②t❤❛❣♦r❡❛♥ t❤❡♦r❡♠✳ ❚❤❡r❡❢♦r❡

(

Ak+1

2k

)2

= sin2360

2k+1=

1− cos 360

2k

2=

1−√

1− sin2 360

2k

2=

1−√

1−(

Ak

2k−1

)2

2

❛♥❞✱ ❝♦♥s❡q✉❡♥t❧②✱

Ak+1 = 2k

√

√

√

√1−√

1−(

Ak

2k−1

)2

2

❲❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♥✉♠❡r✐❝❛❧ ✈❛❧✉❡s✿

A2 = 22sin 360

22

2= 2,

A3 ≈ 2.8284, A4 ≈ 3.0615, A5 ≈ 3.1214, A6 = 3.1365 ❛♥❞ s♦ ♦♥.

❚❤❡♦r❡t✐❝❛❧❧②✿ A2 = 2✱

A3 = 2√2 = 2

2√2, A4 = 2

2 · 2√2 ·

√

2 +√2,

A5 = 22 · 2 · 2

√2 ·

√

2 +√2 ·

√

2 +√

2 +√2

❛♥❞ s♦ ♦♥✱ s❡❡ ❱✐ét❡✬s ❢♦r♠✉❧❛ ✶✳✶✵ ❢♦r ✷✴π✳

✶✸✹ ❈❍❆P❚❊❘ ✶✵✳ ❈■❘❈▲❊❙

❈❤❛♣t❡r ✶✶

❊①❡r❝✐s❡s

✶✶✳✶ ❊①❡r❝✐s❡s

❊①❝❡r❝✐s❡ ✶✶✳✶✳✶ ▲❡t ❛ ❝✐r❝❧❡ ✇✐t❤ r❛❞✐✉s ✷ ❜❡ ❣✐✈❡♥✳ ❚❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥t❤❡ ♣♦✐♥t P ❛♥❞ t❤❡ ❝❡♥t❡r ♦❢ t❤❡ ❝✐r❝❧❡ ✐s 4✳ ❈❛❧❝✉❧❛t❡ t❤❡ ❝♦♠♠♦♥ ❧❡♥❣t❤♦❢ t❤❡ t❛♥❣❡♥t s❡❣♠❡♥ts ❢r♦♠ P t♦ t❤❡ ❣✐✈❡♥ ❝✐r❝❧❡ ❛♥❞ ✜♥❞ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡s❤♦rt❡r ❛r❝ ❛❧♦♥❣ t❤❡ ❝✐r❝❧❡ ❜❡t✇❡❡♥ t❤❡ ❝♦♥t❛❝t ♣♦✐♥ts A ❛♥❞ B✳

❙♦❧✉t✐♦♥✳ ❚❤❡ t❛♥❣❡♥t s❡❣♠❡♥ts ❤❛✈❡ ❛ ❝♦♠♠♦♥ ❧❡♥❣t❤

PA = PB =√42 − 22 =

√12 = 2

√3.

■❢ α ✐s t❤❡ ❝❡♥tr❛❧ ❛♥❣❧❡ ❜❡❧♦♥❣✐♥❣ t♦ t❤❡ s❤♦rt❡r ❛r❝ ❜❡t✇❡❡♥ ❆ ❛♥❞ ❇ t❤❡♥

sinα

2=

√3

2⇒ α = 120.

❚❤❡r❡❢♦r❡120

360=

t❤❡ ❛r❝ ❜❡t✇❡❡♥ ❆ ❛♥❞ ❇2rπ

,

❋✐❣✉r❡ ✶✶✳✶✿ ❊①❡r❝✐s❡ ✶✶✳✶✳✶

✶✸✺

✶✸✻ ❈❍❆P❚❊❘ ✶✶✳ ❊❳❊❘❈■❙❊❙

❋✐❣✉r❡ ✶✶✳✷✿ ❊①❡r❝✐s❡ ✶✶✳✶✳✷

✐✳❡✳

t❤❡ ❛r❝ ❜❡t✇❡❡♥ ❆ ❛♥❞ ❇ =4

3π.

❊①❝❡r❝✐s❡ ✶✶✳✶✳✷ ❚❤❡ r❛❞✐✉s ♦❢ ❛ ❝✐r❝❧❡ ✐s ✶✵✱ t❤❡ t❛♥❣❡♥t ❛t t❤❡ ♣♦✐♥t ❈♦❢ t❤❡ ❝✐r❝❧❡ ❤❛s ❛♥ ✐♥❝❧✐♥❛t✐♦♥ ❛♥❣❧❡ ✸✵ ❞❡❣r❡❡ t♦ t❤❡ ❝❤♦r❞ ❈❇✳ ❖t❤❡r✇✐s❡❆❈ ✐s t❤❡ ❞✐❛♠❡t❡r ♦❢ t❤❡ ❝✐r❝❧❡✳ ❈❛❧❝✉❧❛t❡ t❤❡ ❛r❡❛ ❛♥❞ t❤❡ ♣❡r✐♠❡t❡r ♦❢ t❤❡tr✐❛♥❣❧❡ ❆❇❈✳

❙♦❧✉t✐♦♥✳ ❯s✐♥❣ ❚❤❛❧❡s t❤❡♦r❡♠ ❆❇❈ ✐s ❛ r✐❣❤t tr✐❛♥❣❧❡ ✲ t❤❡ ❛♥❣❧❡ ♦❢ ✾✵❞❡❣r❡❡ ✐♥ ♠❡❛s✉r❡ ✐s s✐t✉❛t❡❞ ❛t ❇✳ ❚❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❤②♣♦t❤❡♥✉s❡ ❆❈ ✐s ✷✵✳❚❤❡ ❛♥❣❧❡ ❛t ❈ ✐s ❥✉st ✻✵ ❜❡❝❛✉s❡ t❤❡ ❝❤♦r❞ ❇❈ ❤❛s ❛♥ ✐♥❝❧✐♥❛t✐♦♥ ❛♥❣❧❡ ✸✵❞❡❣r❡❡ t♦ t❤❡ t❛♥❣❡♥t ❛t t❤❡ ♣♦✐♥t ❈✳ ❚❤❡ ❧❡❣s ❛r❡

20 cos 60 = 10 ❛♥❞ 20 sin 60 = 10√3.

❚❤❡r❡❢♦r❡ t❤❡ ❛r❡❛ ✐s

A = 50√3 ❛♥❞ P = 20 + 10 + 10 ·

√3.

❊①❝❡r❝✐s❡ ✶✶✳✶✳✸ ▲❡t ❆❇ ❜❡ ❛ ❞✐❛♠❡t❡r ♦❢ ❛ ❝✐r❝❧❡ ♦❢ ✉♥✐t r❛❞✐✉s✳ ▲❡t ❈ ❜❡❛ ♣♦✐♥t ♦❢ t❤❡ t❛♥❣❡♥t t♦ t❤❡ ❝✐r❝❧❡ ❛t ❆ ❢♦r ✇❤✐❝❤ ❆❈ ✐s ♦❢ ❧❡♥❣t❤ 2

√3 ❧♦♥❣✳

❈❛❧❝✉❧❛t❡ t❤❡ ❛r❡❛ ♦❢ t❤❡ ❝♦♠♠♦♥ ♣❛rt ♦❢ t❤❡ tr✐❛♥❣❧❡ ❆❇❈ ❛♥❞ t❤❡ ❝✐r❝❧❡✳

❙♦❧✉t✐♦♥✳ ❚❤❡ ❛♥❣❧❡ ❛t ❇ ❝❛♥ ❜❡ ❡❛s✐❧② ❝❛❧❝✉❧❛t❡❞ ❢r♦♠ t❤❡ ❢♦r♠✉❧❛

tan β =AC

AB=

2√3

2⇒ β = 60◦

❚❤❡r❡❢♦r❡ t❤❡ ❝❡♥tr❛❧ ❛♥❣❧❡ ❧②✐♥❣ ♦♥ t❤❡ s❛♠❡ ❛r❝ ✐s ♦❢ ❞❡❣r❡❡ ✶✷✵ ✐♥ ♠❡❛s✉r❡✳❲❡ ❤❛✈❡ t❤❛t

t❤❡ ❛r❡❛ ♦❢ t❤❡ ❝♦♠♠♦♥ ♣❛rt =1

3r2π + t❤❡ ❛r❡❛ ♦❢ ❖❇❉.

✶✶✳✶✳ ❊❳❊❘❈■❙❊❙ ✶✸✼

❋✐❣✉r❡ ✶✶✳✸✿ ❊①❡r❝✐s❡ ✶✶✳✶✳✸

❋✐❣✉r❡ ✶✶✳✹✿ ❊①❡r❝✐s❡ ✶✶✳✶✳✹

❚❤❡ ❛r❡❛ ♦❢ t❤❡ tr✐❛♥❣❧❡ ❖❇❉ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❛s

r2 sin 60

2,

✐✳❡✳

t❤❡ ❛r❡❛ ♦❢ t❤❡ ❝♦♠♠♦♥ ♣❛rt =1

3r2π +

√3

4r2.

❊①❝❡r❝✐s❡ ✶✶✳✶✳✹ ❉r❛✇ ❛ r❤♦♠❜✉s ❛r♦✉♥❞ ❛ ❝✐r❝❧❡ ♦❢ ❛r❡❛ ✶✵✵✱ s♦ t❤❛t t❤❡r❤♦♠❜✉s ❤❛s ❛♥ ❛♥❣❧❡ ✸✵ ❞❡❣r❡❡✳ ❈❛❧❝✉❧❛t❡ t❤❡ ❛r❡❛ ♦❢ t❤❡ r❤♦♠❜✉s✳

❙♦❧✉t✐♦♥✳ ❚❤❡ r❛❞✐✉s ♦❢ t❤❡ ❝✐r❝❧❡ ✐s ✶✵✴√π✳ ■❢ ❛ ✐s t❤❡ ❝♦♠♠♦♥ ❧❡♥❣t❤ ♦❢ t❤❡

s✐❞❡s ♦❢ t❤❡ ❝✐r❝✉♠s❝r✐❜❡❞ r❤♦♠❜✉s t❤❡♥

sin 30 =2r

a⇒ a =

40√π.

✶✸✽ ❈❍❆P❚❊❘ ✶✶✳ ❊❳❊❘❈■❙❊❙

❋✐❣✉r❡ ✶✶✳✺✿ ❊①❡r❝✐s❡ ✶✶✳✶✳✺

❚❤❡r❡❢♦r❡ t❤❡ ❛r❡❛ ♦❢ t❤❡ r❤♦♠❜✉s ✐s

A = a2 sin 30 =800

π.

❊①❝❡r❝✐s❡ ✶✶✳✶✳✺ ❈♦♥str✉❝t ❛♥ ❡q✉✐❧❛t❡r❛❧ tr✐❛♥❣❧❡ ❛❜♦✈❡ t❤❡ ❞✐❛♠❡t❡r ♦❢ ❛❝✐r❝❧❡ ✇✐t❤ r❛❞✐✉s r✳ ❲❤❛t ✐s t❤❡ ❛r❡❛ ♦❢ t❤❡ tr✐❛♥❣❧❡ ❧②✐♥❣ ♦✉ts✐❞❡ t❤❡ ❝✐r❝❧❡✳

❙♦❧✉t✐♦♥✳ ▲❡t ❆❇ ❜❡ t❤❡ ❞✐❛♠❡t❡r ♦❢ t❤❡ ❝✐r❝❧❡ ❛♥❞ ❝♦♥s✐❞❡r t❤❡ ❝♦♠♠♦♥♣♦✐♥ts ❆✬ ❛♥❞ ❇✬ ♦♥ t❤❡ ♣❡r✐♠❡t❡r ♦❢ t❤❡ ❝✐r❝❧❡s✳ ❙✐♥❝❡ ❖❆❆✬ ❛♥❞ ❖❇❇✬ ❛r❡❡q✉✐❧❛t❡r❛❧ tr✐❛♥❣❧❡s ✇✐t❤ s✐❞❡s ♦❢ ❧❡♥❣t❤ r ✐t ❢♦❧❧♦✇s t❤❛t t❤❡ ❛r❡❛ ♦✉ts✐❞❡❢r♦♠ t❤❡ ❝✐r❝❧❡ ✐s ❥✉st

2r · 2r · sin 602

− 2r · r · sin 60

2− 1

6r2π = r2

√3− r2

√3

2− r2

π

6.

❊①❝❡r❝✐s❡ ✶✶✳✶✳✻ ❆ ❝✐r❝❧❡ ♦❢ ✉♥✐t r❛❞✐✉s t♦✉❝❤❡s t❤❡ ❧❡❣s ♦❢ ❛ r✐❣❤t ❛♥❣❧❡✳❲❤❛t ❛r❡ t❤❡ r❛❞✐✐ ♦❢ t❤❡ ❝✐r❝❧❡s ✇❤✐❝❤ t♦✉❝❤❡s t❤❡ t✇♦ ❧❡❣s ♦❢ t❤❡ r✐❣❤t ❛♥❣❧❡❛♥❞ t❤❡ ❣✐✈❡♥ ❝✐r❝❧❡✳

❙♦❧✉t✐♦♥ ✭❝❢✳ ❡①❡r❝✐s❡ ✽✳✶✳✹✮✳ ❉✐✈✐❞❡ t❤❡ ♣r♦❜❧❡♠ ✐♥t♦ t✇♦ ♣❛rts✳ ❆t ✜rst ❧❡t✉s ❝♦♥❝❡♥tr❛t❡ ♦♥ t❤❡ ❝✐r❝❧❡s t♦✉❝❤✐♥❣ t❤❡ ❧❡❣s ♦❢ ❛ r✐❣❤t ❛♥❣❧❡✳ ❚❤❡ ❝❡♥t❡r♦❢ s✉❝❤ ❛ ❝✐r❝❧❡ ♠✉st ❜❡ ♦♥ t❤❡ ❜✐s❡❝t♦r ♦❢ t❤❡ ❛♥❣❧❡✳ ▲❡t ① ❜❡ t❤❡ ❞✐st❛♥❝❡♦❢ t❤❡ ❝❡♥t❡r ❢r♦♠ t❤❡ ✈❡rt❡①✳ P②t❤❛❣♦r❡❛♥ t❤❡♦r❡♠ s❛②s t❤❛t

x2 = r2x + r2x, ✐✳❡✳ x = rx√2,

✇❤❡r❡ rx ✐s t❤❡ r❛❞✐✉s ♦❢ t❤❡ ❝✐r❝❧❡✳ ■t ✐s ❧❛❜❡❧❧❡❞ ❜② t❤❡ ❝♦♦r❞✐♥❛t❡ ①✳ ■♥ ❝❛s❡♦❢ rx = 1 ✇❡ ❤❛✈❡ t❤❛t ①❂

√2✳ ❚✇♦ ❝✐r❝❧❡s ❛r❡ t❛♥❣❡♥t t♦ ❡❛❝❤ ♦t❤❡r ❢r♦♠

♦✉ts✐❞❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❞✐st❛♥❝❡ ♦❢ t❤❡ ❝❡♥t❡rs ✐s t❤❡ s✉♠ ♦❢ t❤❡ r❛❞✐✐✿

|x−√2| = rx + 1,

✶✶✳✶✳ ❊❳❊❘❈■❙❊❙ ✶✸✾

❋✐❣✉r❡ ✶✶✳✻✿ ❊①❡r❝✐s❡ ✶✶✳✶✳✼

✐✳❡✳x−

√2 = rx + 1 ♦r

√2− x = rx + 1.

❚❤❡r❡❢♦r❡

rx =1 +

√2√

2− 1♦r rx =

√2− 1

1 +√2.

❊①❝❡r❝✐s❡ ✶✶✳✶✳✼ ❉r❛✇ ❛ ❝✐r❝❧❡ ❛r♦✉♥❞ t❤❡ ✈❡rt❡① ♦❢ ❛♥ ❛♥❣❧❡ ♦❢ ✶✷✵ ❞❡❣r❡❡✐♥ ♠❡❛s✉r❡✳ ❈❛❧❝✉❧❛t❡ t❤❡ r❛❞✐✉s ♦❢ t❤❡ ❝✐r❝❧❡ ✇❤✐❝❤ t♦✉❝❤❡s t❤❡ ❣✐✈❡♥ ❝✐r❝❧❡✐♥s✐❞❡✱ ❛♥❞ t❤❡ ❧❡❣s ♦❢ t❤❡ ❛♥❣❧❡✳

❙♦❧✉t✐♦♥✳ ▲❡t ❘ ❜❡ t❤❡ r❛❞✐✉s ♦❢ t❤❡ ❝✐r❝❧❡ ❞r❛✇♥ ❛r♦✉♥❞ t❤❡ ✈❡rt❡① ❖ ♦❢ ❛♥❛♥❣❧❡ ♦❢ ✶✷✵ ❞❡❣r❡❡ ✐♥ ♠❡❛s✉r❡✳ ■❢ ❆ ❛♥❞ ❇ ❞❡♥♦t❡ t❤❡ ♣♦✐♥ts ♦❢ t❛♥❣❡♥❝②♦♥ t❤❡ ❧❡❣s ♦❢ t❤❡ ❛♥❣❧❡ t❤❡♥ ❆❇❂r✱ ✇❤❡r❡ r ✐s t❤❡ r❛❞✐✉s ♦❢ t❤❡ ❝✐r❝❧❡ ✇❤✐❝❤t♦✉❝❤❡s t❤❡ ❣✐✈❡♥ ❝✐r❝❧❡ ✐♥s✐❞❡ ❛♥❞ t❤❡ ❧❡❣s ♦❢ t❤❡ ❛♥❣❧❡✳ ❋r♦♠ P②t❤❛❣♦r❡❛♥t❤❡♦r❡♠

R− r =√r2 +OA2,

✇❤❡r❡

OA = r tan 30 ⇒ OA2 =r2

3.

❚❤❡r❡❢♦r❡

R = r

(

1 +2√3

)

⇒ r =R

1 + 2√3

.

❊①❝❡r❝✐s❡ ✶✶✳✶✳✽ ❚❤r❡❡ s✐❞❡s ♦❢ ❛ tr✐❛♥❣❧❡ ❛r❡ ✶✸✱ ✶✹ ❛♥❞ ✶✺✳ ❲❤❛t ✐s t❤❡r❛❞✐✉s ♦❢ t❤❡ ❝✐r❝❧❡ ✇❤♦s❡ ❝❡♥t❡r ❧✐❡s ♦♥ t❤❡ ❧♦♥❣❡st s✐❞❡ ♦❢ t❤❡ tr✐❛♥❣❧❡ ❛♥❞t♦✉❝❤❡s t❤❡ ♦t❤❡r s✐❞❡s✳

❙♦❧✉t✐♦♥✳ ❈♦♥s✐❞❡r t❤❡ r❛❞✐✐ ♦❢ t❤❡ ❝✐r❝❧❡ ✇❤✐❝❤ ❛r❡ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ s✐❞❡s♦❢ ❧❡♥❣t❤s ✶✸ ❛♥❞ ✶✹✱ r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡ ❛r❡❛ ♦❢ t❤❡ tr✐❛♥❣❧❡ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞❛s t❤❡ s✉♠

A =13r

2+

14r

2.

✶✹✵ ❈❍❆P❚❊❘ ✶✶✳ ❊❳❊❘❈■❙❊❙

❋✐❣✉r❡ ✶✶✳✼✿ ❊①❡r❝✐s❡ ✶✶✳✶✳✽

❋✐❣✉r❡ ✶✶✳✽✿ ❊①❡r❝✐s❡ ✶✶✳✶✳✾

❖♥ t❤❡ ♦t❤❡r ❤❛♥❞

A =√

42(42− 13)(42− 14)(42− 15)

❜❡❝❛✉s❡ ♦❢ ❍ér♦♥✬s ❢♦r♠✉❧❛✳ ❋✐♥❛❧❧②

r = 2

√

42(42− 13)(42− 14)(42− 15)

27.

❊①❝❡r❝✐s❡ ✶✶✳✶✳✾ ▲❡t ❘ ❛♥❞ r ❞❡♥♦t❡ t❤❡ r❛❞✐✐ ♦❢ t✇♦ ❝✐r❝❧❡s t♦✉❝❤✐♥❣ ❡❛❝❤♦t❤❡r ♦✉ts✐❞❡ ❛♥❞ ❘ ❃ r✳ ❈❛❧❝✉❧❛t❡ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❝♦♠♠♦♥ ✐♥t❡r♥❛❧ t❛♥❣❡♥t❜❡t✇❡❡♥ t❤❡ ❝♦♠♠♦♥ ❡①t❡r♥❛❧ t❛♥❣❡♥ts✳

❙♦❧✉t✐♦♥✳ ❇❡❝❛✉s❡ ♦❢ t❤❡ s②♠♠❡tr② ✐t ✐s ❡♥♦✉❣❤ t♦ ❝♦♠♣✉t❡ t❤❡ ❤❛❧❢ ♦❢ t❤❡✐♥t❡r♥❛❧ ❝♦♠♠♦♥ t❛♥❣❡♥t✳ ■❢ ❚ ✐s t❤❡ ♣♦✐♥t ♦❢ t❛♥❣❡♥❝② ♦❢ t❤❡ ❝✐r❝❧❡s ✐t ❢♦❧❧♦✇st❤❛t

CA = CT = CB.

❚❤❡r❡❢♦r❡ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ✐♥t❡r♥❛❧ ❝♦♠♠♦♥ t❛♥❣❡♥t ❜❡t✇❡❡♥ t❤❡ ❡①t❡r♥❛❧❝♦♠♠♦♥ t❛♥❣❡♥ts ✐s ❥✉st ❆❇✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞

AB2 + (R− r)2 = (R + r)2

✶✶✳✶✳ ❊❳❊❘❈■❙❊❙ ✶✹✶

❛♥❞✱ ❝♦♥s❡q✉❡♥t❧②✱AB = 2

√Rr

❊①❝❡r❝✐s❡ ✶✶✳✶✳✶✵ ❚❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ s❤♦rt❡st ❞✐❛❣♦♥❛❧ ♦❢ ❛ r❡❣✉❧❛r ✽✕❣♦♥ ✐s❣✐✈❡♥✿ ✶✵✳ ❲❤❛t ✐s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ s✐❞❡s ❛♥❞ t❤❡ ❛r❡❛ ♦❢ t❤❡ ♣♦❧②❣♦♥✳

❙♦❧✉t✐♦♥✳ ▲❡t P1, P2, . . . , P8 ❜❡ t❤❡ ✈❡rt✐❝❡s ♦❢ ❛ r❡❣✉❧❛r ✽✕❣♦♥ ✐♥s❝r✐❜❡❞ ✐♥ ❛❝✐r❝❧❡ ✇✐t❤ ❝❡♥t❡r ❖✳ ❚❤❡ s❤♦rt❡st ❞✐❛❣♦♥❛❧ ❝♦♥♥❡❝t✐♥❣ P1 ❛♥❞ P3 ❜❡❧♦♥❣s t♦t❤❡ ❝❡♥tr❛❧ ❛♥❣❧❡ ♦❢ ✾✵ ❞❡❣r❡❡ ✐♥ ♠❡❛s✉r❡ ❜❡❝❛✉s❡

6 P1OP3 = 2 · 6 P1OP2 = 2360

8= 90.

❯s✐♥❣ P②t❤❛❣♦r❡❛♥ t❤❡♦r❡♠ ✐t ❢♦❧❧♦✇s t❤❛t

r2 + r2 = 100 ⇒ r =√50.

❚❤❡r❡❢♦r❡

P1P2 =√r2 + r2 − 2 · r · r · cos 45 =

√50 + 50− 2 · 50 · 50 · cos 45

❛♥❞ t❤❡ ❛r❡❛ ✐s

A = 8r · r · sin 45

2.

❊①❝❡r❝✐s❡ ✶✶✳✶✳✶✶ ❆❧❧ s✐❞❡s ♦❢ ❛ s②♠♠❡tr✐❝❛❧ tr❛♣❡③♦✐❞ t♦✉❝❤ ❛ ❝✐r❝❧❡✳ ❚❤❡♣❛r❛❧❧❡❧ ❜❛s❡s ❛r❡ ✶✵ ❛♥❞ ✷✵✳

• ❈❛❧❝✉❧❛t❡ t❤❡ ❛♥❣❧❡s ♦❢ t❤❡ tr❛♣❡③♦✐❞✳

• ❈❛❧❝✉❧❛t❡ t❤❡ ❛r❡❛ ♦❢ t❤❡ tr❛♣❡③♦✐❞✳

❊①❝❡r❝✐s❡ ✶✶✳✶✳✶✷ ❲❤❛t ❛r❡ t❤❡ ❛♥❣❧❡s ♦❢ ❛ r❤♦♠❜✉s ✐❢ ✐ts ❛r❡❛ ✐s ❥✉st t✇✐❝❡♦❢ t❤❡ ❛r❡❛ ♦❢ t❤❡ ✐♥s❝r✐❜❡❞ ❝✐r❝❧❡❄

❊①❝❡r❝✐s❡ ✶✶✳✶✳✶✸ ❚❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ s❤♦rt❡st ❞✐❛❣♦♥❛❧ ♦❢ ❛ r❡❣✉❧❛r ✽✕❣♦♥ ✐s❣✐✈❡♥✿ ✷✵✳

• ❲❤❛t ✐s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ s✐❞❡s❄

• ❲❤❛t ✐s t❤❡ ❛r❡❛ ♦❢ t❤❡ ♣♦❧②❣♦♥❄

❊①❝❡r❝✐s❡ ✶✶✳✶✳✶✹ ❚❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ s✐❞❡ ♦❢ ❛ r❡❣✉❧❛r ✻✕❣♦♥ ✐s ❣✐✈❡♥✿ ✽✳

• ❈❛❧❝✉❧❛t❡ t❤❡ ❛♥❣❧❡s ♦❢ t❤❡ ♣♦❧②❣♦♥✳

✶✹✷ ❈❍❆P❚❊❘ ✶✶✳ ❊❳❊❘❈■❙❊❙

❋✐❣✉r❡ ✶✶✳✾✿ ❊①❡r❝✐s❡ ✶✶✳✶✳✶✻

• ❲❤❛t ✐s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ s❤♦rt❡st ❞✐❛❣♦♥❛❧❄

• ❲❤❛t ✐s t❤❡ ❛r❡❛ ♦❢ t❤❡ ♣♦❧②❣♦♥❄

❊①❝❡r❝✐s❡ ✶✶✳✶✳✶✺ ❚✇♦ ❝✐r❝❧❡s ♦❢ r❛❞✐✉s ✺ ✐♥t❡rs❡❝t ❡❛❝❤ ♦t❤❡r✳ ❚❤❡ ❞✐s✲t❛♥❝❡ ♦❢ t❤❡ t❤❡✐r ❝❡♥t❡rs ✐s ✽✳ ❈❛❧❝✉❧❛t❡ t❤❡ ❛r❡❛ ♦❢ t❤❡ ❝♦♠♠♦♥ ♣❛rt ♦❢ t❤❡❝✐r❝❧❡s✳

❊①❝❡r❝✐s❡ ✶✶✳✶✳✶✻ ❆ ♣♦❧②❣♦♥ ♦❢ ✶✷ s✐❞❡s ❝❛♥ ❜❡ ✐♥s❝r✐❜❡❞ ✐♥t♦ ❛ ❝✐r❝❧❡✳ ❙✐①♦❢ t❤❡ s✐❞❡s ❤❛✈❡ ❧❡♥❣t❤

√2✱ ❛♥❞ t❤❡ ♦t❤❡r s✐① s✐❞❡s ❛r❡ ❡q✉❛❧ t♦

√24✳ ❲❤❛t

✐s t❤❡ r❛❞✐✉s ♦❢ t❤❡ ❝✐r❝❧❡✳

❙♦❧✉t✐♦♥✳ ▲❡t ❖ ❜❡ t❤❡ ❝❡♥t❡r ♦❢ t❤❡ ❝✐r❝❧❡ ❛♥❞ ❝♦♥s✐❞❡r t❤❡ ✈❡rt✐❝❡s ❆✱ ❇❛♥❞ ❈ ♦❢ t❤❡ ♣♦❧②❣♦♥ s✉❝❤ t❤❛t

AB =√2 ❛♥❞ BC =

√24.

■❢ α ❛♥❞ β ❞❡♥♦t❡ t❤❡ ❝❡♥tr❛❧ ❛♥❣❧❡s ❜❡❧♦♥❣✐♥❣ t♦ ❆❇ ❛♥❞ ❇❈✱ r❡s♣❡❝t✐✈❡❧②✇❡ ❤❛✈❡ t❤❛t

6α + 6β = 360

❛♥❞✱ ❝♦♥s❡q✉❡♥t❧②α + β = 60.

❚❤❡r❡❢♦r❡ t❤❡ tr✐❛♥❣❧❡ ❆❖❈ ✐s ❡q✉✐❧❛t❡r❛❧ ❛♥❞ ❆❈ ❂ r✳ ❚♦ ✜♥✐s❤ t❤❡ s♦❧✉t✐♦♥✇❡ ❝♦♠♣✉t❡ t❤❡ ❛♥❣❧❡ ❛t ❇ ✐♥ t❤❡ tr✐❛♥❣❧❡ ❆❇❈✳ ❈❤♦♦s❡ t❤❡ ♣♦✐♥t ❇✬ ♦♥ t❤❡❝✐r❝❧❡ ♦♣♣♦s✐t❡ t♦ ❇✳ ❚❤❡♥ ❇✬❆❇❈ ❢♦r♠ ❛ ❝②❝❧✐❝ q✉❛❞r✐❧❛t❡r❛❧ ❛♥❞ t❤❡ s✉♠ ♦❢

✶✶✳✶✳ ❊❳❊❘❈■❙❊❙ ✶✹✸

t❤❡ ♠❡❛s✉r❡s ♦❢ t❤❡ ♦♣♣♦s✐t❡ ❛♥❣❧❡s ♠✉st ❜❡ ✶✽✵ ❞❡❣r❡❡✳ ❚❤❡ ✐♥s❝r✐❜❡❞ ❛♥❣❧❡t❤❡♦r❡♠ s❛②s t❤❛t

6 AB′C = 30

❛♥❞✱ ❝♦♥s❡q✉❡♥t❧②✱

6 ABC = 180− 6 AB′C = 150.

❯s✐♥❣ t❤❡ ❝♦s✐♥❡ r✉❧❡ ✇❡ ❤❛✈❡ t❤❛t

AC2 = 24 + 2− 2 · 24 · 2 · cos 150 = r2.

✶✹✹ ❈❍❆P❚❊❘ ✶✶✳ ❊❳❊❘❈■❙❊❙

❈❤❛♣t❡r ✶✷

●❡♦♠❡tr✐❝ tr❛♥s❢♦r♠❛t✐♦♥s

✶✷✳✶ ■s♦♠❡tr✐❡s

❉❡✜♥✐t✐♦♥ ❚❤❡ ♣♦✐♥t tr❛♥s❢♦r♠❛t✐♦♥ ϕ : P → P ′ ✐s ❝❛❧❧❡❞ ❛♥ ✐s♦♠❡tr② ✐❢ ✐t♣r❡s❡r✈❡s t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ♣♦✐♥ts✿

PQ = P ′Q′.

❆❝❝♦r❞✐♥❣ t♦ t❤❡ ❝❛s❡ ❙❙❙ ♦❢ t❤❡ ❝♦♥❣r✉❡♥❝❡ ♦❢ tr✐❛♥❣❧❡s ❛♥② ✐s♦♠❡tr②♣r❡s❡r✈❡s t❤❡ ❛♥❣❧❡s ❛♥❞✱ ❜② t❤❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ♣❛r❛❧❧❡❧✐s♠✱ t❤❡ ♣❛r❛❧✲❧❡❧✐s♠✿ ♣❛r❛❧❧❡❧ ❧✐♥❡s ❛r❡ tr❛♥s❢♦r♠❡❞ ✐♥t♦ ♣❛r❛❧❧❡❧ ❧✐♥❡s ✉♥❞❡r ❛♥② ✐s♦♠❡tr②✳■♥ ✇❤❛t ❢♦❧❧♦✇s ✇❡ ❝❧❛ss✐❢② t❤❡ ♣♦ss✐❜❧❡ ❝❛s❡s ✐♥ t❡r♠s ♦❢ t❤❡ ✜①♣♦✐♥ts✳

❚❤❡♦r❡♠ ✶✷✳✶✳✶ ■❢ ❛♥ ✐s♦♠❡tr② ❤❛s t✇♦ ✜①♣♦✐♥ts ❆ ❛♥❞ ❇ t❤❡♥ ❢♦r ❛♥②♣♦✐♥t ❳ ♦❢ t❤❡ ❧✐♥❡ ❆❇ ✇❡ ❤❛✈❡

X ′ = X.

Pr♦♦❢ ❙✐♥❝❡ ❆✬❂❆ ❛♥❞ ❇✬❂❇ ✇❡ ❤❛✈❡ t❤❛t

AX = A′X ′ = AX ′ ❛♥❞ BX = B′X ′ = BX ′.

❚❤❡r❡❢♦r❡ ❳✬ ♠✉st ❜❡ ♦♥

• t❤❡ ❝✐r❝❧❡ ❛r♦✉♥❞ ❆ ✇✐t❤ r❛❞✐✉s ❆❳✱

• t❤❡ ❝✐r❝❧❡ ❛r♦✉♥❞ ❇ ✇✐t❤ r❛❞✐✉s ❇❳✳

❙✐♥❝❡ ❆✱ ❇ ❛♥❞ ❳ ❛r❡ ❝♦❧❧✐♥❡❛r ♣♦✐♥ts t❤❡s❡ ❝✐r❝❧❡s ❛r❡ t❛♥❣❡♥t t♦ ❡❛❝❤ ♦t❤❡r❛t t❤❡ ✉♥✐q✉❡❧② ❞❡t❡r♠✐♥❡❞ ♣♦✐♥t ❳ ♦❢ t❛♥❣❡♥❝②✿ ❳✬❂❳✳

✶✹✺

✶✹✻ ❈❍❆P❚❊❘ ✶✷✳ ●❊❖▼❊❚❘■❈ ❚❘❆◆❙❋❖❘▼❆❚■❖◆❙

❈♦r♦❧❧❛r② ✶✷✳✶✳✷ ■❢ ❛♥ ✐s♦♠❡tr② ❤❛s t❤r❡❡ ♥♦t ❝♦❧❧✐♥❡❛r ✜①♣♦✐♥ts t❤❡♥ ✐t♠✉st ❜❡ t❤❡ ✐❞❡♥t✐t②✳

Pr♦♦❢ ❙✉♣♣♦s❡ t❤❛t ❆✱ ❇ ❛♥❞ ❈ ❛r❡ ♥♦t ❝♦❧❧✐♥❡❛r ✜①♣♦✐♥ts✳ ▲❡t ❉ ❜❡ ❛♥❛r❜✐tr❛r② ❡❧❡♠❡♥t ✐♥ t❤❡ ♣❧❛♥❡ ❛♥❞ ❝♦♥s✐❞❡r t❤❡ ♣❛r❛❧❧❡❧ ❧✐♥❡ t♦ ❇❈ ♣❛ss✐♥❣t❤r♦✉❣❤ ❉✳ ❚❤✐s ❧✐♥❡ ✐♥t❡rs❡❝t ❜♦t❤ ❆❇ ❛♥❞ ❆❈ ❛t t❤❡ ♣♦✐♥ts ❋ ❛♥❞ ●✱r❡s♣❡❝t✐✈❡❧②✳ ❇② t❤❡♦r❡♠ ✶✷✳✶✳✶ ✐t ❢♦❧❧♦✇s t❤❛t ❋✬❂❋✱ ●✬❂● ❛♥❞ ❉✬❂❉✳

❚❤❡ ❝❛s❡ ♦❢ t✇♦ ✜①♣♦✐♥ts ❣✐✈❡s t❤❡ ✐❞❡♥t✐❝❛❧ tr❛♥s❢♦r♠❛t✐♦♥ ♦r t❤❡ r❡✢❡❝✲t✐♦♥ ❛❜♦✉t t❤❡ ❧✐♥❡ ✭❛①✐s✮ ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ✜①♣♦✐♥ts ❆ ❛♥❞ ❇✳ ■♥ t❤❡ s❡♥s❡♦❢ t❤❡♦r❡♠ ✶✷✳✶✳✶ ❢♦r ❛♥② ❡❧❡♠❡♥t ♦❢ t❤❡ ❧✐♥❡ ❆❇ ✇❡ ❤❛✈❡ t❤❛t ❳✬❂❳✳ ❲❤❛t❛❜♦✉t t❤❡ ♣♦✐♥ts ♥♦t ✐♥ t❤❡ ❛①✐s ♦❢ t❤❡ r❡✢❡❝t✐♦♥❄ ▲❡t ❨ ❜❡ ♦♥❡ ♦❢ t❤❡♠✳❙✐♥❝❡

Y A = Y ′A′ = Y ′A

❛♥❞Y B = Y ′B′ = Y ′B

✐t ❢♦❧❧♦✇s ❜② t❤❡ ❣❡♦♠❡tr✐❝ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ t❤❡ ♣❡r♣❡♥❞✐❝✉❧❛r ❜✐s❡❝t♦r t❤❛tt❤❡ ❧✐♥❡ ❆❇ ✐s ❥✉st t❤❡ ♣❡r♣❡♥❞✐❝✉❧❛r ❜✐s❡❝t♦r ♦❢ t❤❡ s❡❣♠❡♥t ❨❨✬✳❚❤❡ ❝❛s❡ ♦❢ ❡①❛❝t❧② ♦♥❡ ✜①♣♦✐♥t r❡s✉❧ts ✐♥ t❤❡ ♥♦t✐♦♥ ♦❢ r♦t❛t✐♦♥ ❛❜♦✉tt❤❡ ✉♥✐q✉❡❧② ❞❡t❡r♠✐♥❡❞ ✜①♣♦✐♥t ❖✳❚r❛♥s❧❛t✐♦♥s ❛r❡ t②♣✐❝❛❧ ❡①❛♠♣❧❡s ♦♥ ✐s♦♠❡tr✐❡s ✇✐t❤♦✉t ✜①♣♦✐♥ts✳

❉❡✜♥✐t✐♦♥ ❚✇♦ s✉❜s❡ts ✐♥ t❤❡ ♣❧❛♥❡ ❛r❡ ❝❛❧❧❡❞ ❝♦♥❣r✉❡♥t ✐❢ t❤❡r❡ ✐s ❛♥ ✐s♦♠✲❡tr② ✇❤✐❝❤ tr❛♥s❢♦r♠ t❤❡♠ ✐♥t♦ ❡❛❝❤ ♦t❤❡r✳

❆❝❝♦r❞✐♥❣ t♦ t❤❡ ♣r✐♥❝✐♣❧❡ ♦❢ ♣❡r♠❛♥❡♥❝❡ ✇❡ s❤♦✉❧❞ ❝❤❡❝❦ t❤❛t ✐♥ ❝❛s❡♦❢ t✇♦ ❝♦♥❣r✉❡♥t tr✐❛♥❣❧❡s ❆❇❈ ❛♥❞ ❉❊❋ t❤❡r❡ ✐s ❛♥ ✐s♦♠❡tr② ✇❤✐❝❤ ♠❛♣s❆❇❈ ✐♥t♦ ❉❊❋✳ ❚❤❡ ❜❛s✐❝ st❡♣s ♦❢ t❤❡ ❝♦♥str✉❝t✐♦♥ ❝❛♥ ❜❡ ❢♦r♠✉❧❛t❡❞ ❛s❢♦❧❧♦✇s✿

• ■❢ ❆❂❉ t❤❡♥ ✇❡ ✉s❡ t❤❡ ✐❞❡♥t✐❝❛❧ tr❛♥s❢♦r♠❛t✐♦♥ ❛s t❤❡ ✜rst✳ ❖t❤❡r✲✇✐s❡ r❡✢❡❝t t❤❡ tr✐❛♥❣❧❡ ❆❇❈ ❛❜♦✉t t❤❡ ♣❡r♣❡♥❞✐❝✉❧❛r ❜✐s❡❝t♦r ♦❢ t❤❡s❡❣♠❡♥t ❆❉✳ ❚❤✐s r❡s✉❧ts ✐♥ ❛ tr✐❛♥❣❧❡ ❆✬❇✬❈✬✱ ✇❤❡r❡ ❆✬❂❉

• ■❢ ❇✬❂❊ t❤❡♥ ✇❡ ✉s❡ t❤❡ ✐❞❡♥t✐❝❛❧ tr❛♥s❢♦r♠❛t✐♦♥ ❛s t❤❡ s❡❝♦♥❞✳ ❖t❤✲❡r✇✐s❡ r❡✢❡❝t t❤❡ tr✐❛♥❣❧❡ ❆✬❇✬❈✬ ❛❜♦✉t t❤❡ ♣❡r♣❡♥❞✐❝✉❧❛r ❜✐s❡❝t♦r ♦❢t❤❡ s❡❣♠❡♥t ❇✬❊✳ ❚❤✐s r❡s✉❧ts ✐♥ ❛ tr✐❛♥❣❧❡ ❆✑❇✑❈✑✱ ✇❤❡r❡ ❇✑❂❊✳ ❲❤❛t❛❜♦✉t ❆✑ ❄ ❚♦ ❛♥s✇❡r t❤❡ q✉❡st✐♦♥ ✇❡ s❤♦✉❧❞ ❝♦♠♣❛r❡ t❤❡ ❞✐st❛♥❝❡s❆✬❇✬ ❛♥❞ ❆✬❊✿

A′B′ = AB = DE

✶✷✳✶✳ ■❙❖▼❊❚❘■❊❙ ✶✹✼

❜❡❝❛✉s❡ t❤❡ tr✐❛♥❣❧❡s ❆❇❈ ❛♥❞ ❉❊❋ ❛r❡ ❝♦♥❣r✉❡♥t✳ ❯s✐♥❣ t❤❡ ✜rst st❡♣

DE = A′E ⇒ A′B′ = A′E.

❚❤❡ ❣❡♦♠❡tr✐❝ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ t❤❡ ♣❡r♣❡♥❞✐❝✉❧❛r ❜✐s❡❝t♦r ✐♠♣❧✐❡st❤❛t ❆✬ ✐s ❧②✐♥❣ ♦♥ t❤❡ ❛①✐s ♦❢ t❤❡ r❡✢❡❝t✐♦♥✳ ❚❤❡r❡❢♦r❡

A′′ = (A′)′ = A′ = D.

• ■❢ ❈✑❂❋ t❤❡♥ ✇❡ ✉s❡ t❤❡ ✐❞❡♥t✐❝❛❧ tr❛♥s❢♦r♠❛t✐♦♥ ❛s t❤❡ t❤✐r❞✳ ❖t❤✲❡r✇✐s❡ r❡✢❡❝t t❤❡ tr✐❛♥❣❧❡ ❆✑❇✑❈✑ ❛❜♦✉t t❤❡ ♣❡r♣❡♥❞✐❝✉❧❛r ❜✐s❡❝t♦r ♦❢t❤❡ s❡❣♠❡♥t ❈✑❋✳ ❚❤✐s r❡s✉❧ts ✐♥ ❛ tr✐❛♥❣❧❡ ❆✑ ✬❇✑ ✬❈✑ ✬✱ ✇❤❡r❡ ❈✑ ✬❂❋✳❲❤❛t ❛❜♦✉t ❆✑✬ ❛♥❞ ❇✑✬❄

❲❡ ❛r❡ ❣♦✐♥❣ t♦ ♣r♦✈❡ t❤❛t

A′′′ = A′′ = D ❛♥❞ B′′′ = B′′ = E.

■♥ ♦r❞❡r t♦ ❝❤❡❝❦ t❤❡ ✜rst st❛t❡♠❡♥t ✇❡ s❤♦✉❧❞ ❝♦♠♣❛r❡ t❤❡ ❞✐st❛♥❝❡s ❆✑❈✑❛♥❞ ❆✑❋✿

A′′C ′′ = A′C ′ = AC = DF

❜❡❝❛✉s❡ t❤❡ tr✐❛♥❣❧❡s ❆❇❈ ❛♥❞ ❉❊❋ ❛r❡ ❝♦♥❣r✉❡♥t✳ ❯s✐♥❣ t❤❡ ✜rst ❛♥❞ t❤❡s❡❝♦♥❞ st❡♣s

DF = A′F = A′′F ⇒ A′′C ′′ = A′′F.

❚❤❡ ❣❡♦♠❡tr✐❝ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ t❤❡ ♣❡r♣❡♥❞✐❝✉❧❛r ❜✐s❡❝t♦r ✐♠♣❧✐❡s t❤❛t ❆✑✐s ❧②✐♥❣ ♦♥ t❤❡ ❛①✐s ♦❢ t❤❡ r❡✢❡❝t✐♦♥✳ ❚❤❡r❡❢♦r❡

A′′′ = (A′′)′ = A′′ = D.

❚❤❡ ♣r♦♦❢ ♦❢ t❤❡ s❡❝♦♥❞ st❛t❡♠❡♥t ✐s s✐♠✐❧❛r✿

B′′C ′′ = B′C ′ = BC = EF

❜❡❝❛✉s❡ t❤❡ tr✐❛♥❣❧❡s ❆❇❈ ❛♥❞ ❉❊❋ ❛r❡ ❝♦♥❣r✉❡♥t✳ ❯s✐♥❣ t❤❡ s❡❝♦♥❞ st❡♣

EF = B′′F ⇒ B′′C ′′ = B′′F.

❚❤❡ ❣❡♦♠❡tr✐❝ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ t❤❡ ♣❡r♣❡♥❞✐❝✉❧❛r ❜✐s❡❝t♦r ✐♠♣❧✐❡s t❤❛t ❇✑✐s ❧②✐♥❣ ♦♥ t❤❡ ❛①✐s ♦❢ t❤❡ r❡✢❡❝t✐♦♥✳ ❚❤❡r❡❢♦r❡

B′′′ = (B′′)′ = B′′ = E.

✶✹✽ ❈❍❆P❚❊❘ ✶✷✳ ●❊❖▼❊❚❘■❈ ❚❘❆◆❙❋❖❘▼❆❚■❖◆❙

❋✐❣✉r❡ ✶✷✳✶✿ ❚❤❡ ♣r✐♥❝✐♣❧❡ ♦❢ ♣❡r♠❛♥❡♥❝❡✳

❘❡♠❛r❦ ❚❤❡ ❞✐s❝✉ss✐♦♥ ♦❢ t❤❡ ♣r✐♥❝✐♣❧❡ ♦❢ ♣❡r♠❛♥❡♥❝❡ s❤♦✇s t❤❛t ❢♦r ❛♥②♣❛✐r ♦❢ ❝♦♥❣r✉❡♥t tr✐❛♥❣❧❡s ❆❇❈ ❛♥❞ ❉❊❋ t❤❡r❡ ❡①✐sts ❛♥ ✐s♦♠❡tr② s✉❝❤ t❤❛t

A′ = D, B′ = E ❛♥❞ C ′ = F.

❈♦r♦❧❧❛r② ✶✷✳✶✳✷ ♣r♦✈✐❞❡s t❤❡ ✉♥✐❝✐t② ♦❢ s✉❝❤ ❛♥ ✐s♦♠❡tr② t♦♦✳ ❚❤❡r❡❢♦r❡ ❛♥②✐s♦♠❡tr② ✐s ✉♥✐q✉❡❧② ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ✐♠❛❣❡s ♦❢ t❤r❡❡ ♥♦t ❝♦❧❧✐♥❡❛r ♣♦✐♥ts✳❆t t❤❡ s❛♠❡ t✐♠❡ ❛♥② ✐s♦♠❡tr② ✐s t❤❡ ♣r♦❞✉❝t ♦❢ ❛t ♠♦st t❤r❡❡ r❡✢❡❝t✐♦♥s❛❜♦✉t ❧✐♥❡s✳ ❚❤✐s ❣✐✈❡s ❛ ♥❡✇ st❛rt✐♥❣ ♣♦✐♥t ❢♦r t❤❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥✿ ❛♥②✐s♦♠❡tr② ✐s ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ t②♣❡s

• r❡✢❡❝t✐♦♥ ❛❜♦✉t ❛ ❧✐♥❡✱

• t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ r❡✢❡❝t✐♦♥s ❛❜♦✉t ❧✐♥❡s✱

• t❤❡ ♣r♦❞✉❝t ♦❢ t❤r❡❡ r❡✢❡❝t✐♦♥s ❛❜♦✉t ❧✐♥❡s✳

❊①❝❡r❝✐s❡ ✶✷✳✶✳✸ Pr♦✈❡ t❤❛t t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ r❡✢❡❝t✐♦♥s ❛❜♦✉t ❧✐♥❡s ✐s ❛r♦t❛t✐♦♥ ♦r ❛ tr❛♥s❧❛t✐♦♥ ❞❡♣❡♥❞✐♥❣ ♦♥ ✇❤❡t❤❡r t❤❡ ❛①❡s ❛r❡ ❝♦♥❝✉rr❡♥t ♦r♣❛r❛❧❧❡❧✳

✶✷✳✷ ❙✐♠✐❧❛r✐t✐❡s

❉❡✜♥✐t✐♦♥ ❚❤❡ ♣♦✐♥t tr❛♥s❢♦r♠❛t✐♦♥ ξ : P → P ′ ✐s ❝❛❧❧❡❞ ❛ s✐♠✐❧❛r✐t② ✐❢ ✐t♣r❡s❡r✈❡s t❤❡ r❛t✐♦ ♦❢ ❞✐st❛♥❝❡s ❜❡t✇❡❡♥ t❤❡ ♣♦✐♥ts✿

PQ : P ′Q′ = λ,

✇❤❡r❡ t❤❡ ♣♦s✐t✐✈❡ ❝♦♥st❛♥t λ ✐s ❝❛❧❧❡❞ t❤❡ s✐♠✐❧❛r✐t② r❛t✐♦✳

✶✷✳✷✳ ❙■▼■▲❆❘■❚■❊❙ ✶✹✾

❆❝❝♦r❞✐♥❣ t♦ t❤❡ ❝❛s❡ ❙✬❙✬❙✬ ♦❢ t❤❡ s✐♠✐❧❛r✐t② ♦❢ tr✐❛♥❣❧❡s ❛♥② s✐♠✐❧❛r✐t②tr❛♥s❢♦r♠❛t✐♦♥ ♣r❡s❡r✈❡s t❤❡ ❛♥❣❧❡s ❛♥❞✱ ❜② t❤❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ♣❛r❛❧✲❧❡❧✐s♠✱ t❤❡ ♣❛r❛❧❧❡❧✐s♠✿ ♣❛r❛❧❧❡❧ ❧✐♥❡s ❛r❡ tr❛♥s❢♦r♠❡❞ ✐♥t♦ ♣❛r❛❧❧❡❧ ❧✐♥❡s ✉♥❞❡r❛♥② s✐♠✐❧❛r✐t②✳ ■♥ ✇❤❛t ❢♦❧❧♦✇s ✇❡ ❝❧❛ss✐❢② t❤❡ ♣♦ss✐❜❧❡ ❝❛s❡s ✐♥ t❡r♠s ♦❢ t❤❡✜①♣♦✐♥ts✳

❘❡♠❛r❦ ■s♦♠❡tr✐❡s ❛r❡ s✐♠✐❧❛r✐t✐❡s ✇✐t❤ r❛t✐♦ ✶✳

❚❤❡♦r❡♠ ✶✷✳✷✳✶ ■❢ ❛ s✐♠✐❧❛r✐t② ✐s ♥♦t ❛♥ ✐s♦♠❡tr② t❤❡♥ ✐t ❤❛s ❛ ✉♥✐q✉❡❧②❞❡t❡r♠✐♥❡❞ ✜①♣♦✐♥t✳

Pr♦♦❢ ■t ✐s ❝❧❡❛r t❤❛t ✐❢ ✇❡ ❤❛✈❡ t✇♦ ❞✐✛❡r❡♥t ✜①♣♦✐♥ts t❤❡♥ t❤❡ tr❛♥s❢♦r♠❛✲t✐♦♥ ✐s ❛♥ ✐s♦♠❡tr②✳ ❚❤❡r❡❢♦r❡ t❤❡ ♥✉♠❜❡r ♦❢ ✜①♣♦✐♥ts ✐s ❛t ♠♦st ♦♥❡✳ ❋♦rt❤❡ ♣r♦♦❢ ♦❢ t❤❡ ❡①✐st❡♥❝❡ ✇❡ ❝❛♥ r❡❢❡r t♦ ❬✺❪✱ ✇❤❡r❡ ❛♥ ❡❧❡♠❡♥t❛r② r✉❧❡r ❝♦♥✲str✉❝t✐♦♥ ❝❛♥ ❜❡ ❢♦✉♥❞ ❢♦r ✜♥❞✐♥❣ t❤❡ ✜①♣♦✐♥t ♦❢ ❛ s✐♠✐❧❛r✐t② tr❛♥s❢♦r♠❛t✐♦♥✐♥ t❤❡ ♣❧❛♥❡✳

❆♥ ✐♠♣♦rt❛♥t s✉❜❝❧❛ss ♦❢ s✐♠✐❧❛r✐t✐❡s ✐s ❢♦r♠❡❞ ❜② t❤❡ ❝❡♥tr❛❧ s✐♠✐❧❛r✐t✐❡s❀s❡❡ t❤❡ ♣r♦♦❢ ♦❢ t❤❡♦r❡♠ ✹✳✷✳✶✳

❈♦r♦❧❧❛r② ✶✷✳✷✳✷ ❆♥② s✐♠✐❧❛r✐t② ❝❛♥ ❜❡ ❣✐✈❡♥ ❛s t❤❡ ♣r♦❞✉❝t ♦❢ ❛ ❝❡♥tr❛❧s✐♠✐❧❛r✐t② ❛♥❞ ❛♥ ✐s♦♠❡tr②✳

Pr♦♦❢ ▲❡t ξ ❜❡ ❛ s✐♠✐❧❛r✐t② ✇✐t❤ r❛t✐♦ λ✳ ■❢ λ = 1 t❤❡♥ ✇❡ ❤❛✈❡ ❛♥ ✐s♦♠❡tr②✳❖t❤❡r✇✐s❡ t❤❡ ✜①♣♦✐♥t ❈ ♦❢ ξ ✐s ✉♥✐q✉❡❧② ❞❡t❡r♠✐♥❡❞ ✐♥ t❤❡ s❡♥s❡ ♦❢ t❤❡♦r❡♠✶✷✳✷✳✶✳ ❚❤❡r❡❢♦r❡ t❤❡ ♣r♦❞✉❝t ♦❢ ξ ❛♥❞ t❤❡ ❝❡♥tr❛❧ s✐♠✐❧❛r✐t② ♦❢ s❝❛❧✐♥❣ 1/λ✇✐t❤ r❡s♣❡❝t t♦ ❈ ❣✐✈❡s ❛♥ ✐s♦♠❡tr② ϕ✳

❘❡♠❛r❦ ❙✐♥❝❡ ❈ ♠✉st ❜❡ t❤❡ ✜①♣♦✐♥t ♦❢ ϕ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣♦ss✐❜❧❡❝❛s❡s✿ ✐❢ ϕ ✐s

• t❤❡ ✐❞❡♥t✐t② t❤❡♥ ξ ✐s ❛ ❝❡♥tr❛❧ s✐♠✐❧❛r✐t②✱

• ❛ r❡✢❡❝t✐♦♥ ❛❜♦✉t ❛ ❧✐♥❡ ♣❛ss✐♥❣ t❤r♦✉❣❤ ❈ t❤❡♥ ξ ✐s ❛ s♦ ✲ ❝❛❧❧❡❞ str❡t❝❤r❡✢❡❝t✐♦♥✳

• ❛ r♦t❛t✐♦♥ ❛❜♦✉t ❈ t❤❡♥ ξ ✐s ❛ s♦ ✲ ❝❛❧❧❡❞ str❡t❝❤ r♦t❛t✐♦♥ ♦r s♣✐r❛❧s✐♠✐❧❛r✐t②✳

✶✺✵ ❈❍❆P❚❊❘ ✶✷✳ ●❊❖▼❊❚❘■❈ ❚❘❆◆❙❋❖❘▼❆❚■❖◆❙

❈❤❛♣t❡r ✶✸

❈❧❛ss✐❝❛❧ ♣r♦❜❧❡♠s ■■

❊✈❡r②❜♦❞② ❦♥♦✇s t❤❡ ❢❛♠♦✉s ❣❡♦♠❡tr✐❝ ♣r✐♥❝✐♣❧❡ ❛❜♦✉t t❤❡ s❤♦rt❡st ✇❛②❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts✳ ■♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦❜❧❡♠s ✇❡ ❝❛♥ ♥♦t ✉s❡ t❤❡ ♣r✐♥❝✐♣❧❡✐♥ ❛ ❞✐r❡❝t ✇❛② ❜❡❝❛✉s❡ t❤❡ str❛✐❣❤t ❧✐♥❡ s❡❣♠❡♥ts ❛r❡ ❢♦r❜✐❞❞❡♥ ❜② s♦♠❡❝♦♥str❛✐♥ts✳ ❚❤❡ ✐♥❞✐r❡❝t ✇❛② ✐s ❜❛s❡❞ ♦♥ ✉s✐♥❣ ❣❡♦♠❡tr✐❝ tr❛♥s❢♦r♠❛t✐♦♥s t♦❝r❡❛t❡ ❛ ♥❡✇ ❝♦♥✜❣✉r❛t✐♦♥ ❢♦r t❤❡ ❞✐r❡❝t ❛♣♣❧✐❝❛t✐♦♥✳ ❚♦ ❦❡❡♣ ❛❧❧ t❤❡ ♠❡tr✐❝r❡❧❛t✐♦♥s❤✐♣s t❤❡ tr❛♥s❢♦r♠❛t✐♦♥s ♠✉st ❜❡ ✐s♦♠❡tr✐❡s✳

✶✸✳✶ ❚❤❡ ♣r♦❜❧❡♠ ♦❢ t❤❡ ❜r✐❞❣❡

Pr♦❜❧❡♠✳ ❙✉♣♣♦s❡ t❤❛t t❤❡r❡ ❛r❡ t✇♦ ✈✐❧❧❛❣❡s ❆ ❛♥❞ ❇ ♦♥ ❞✐✛❡r❡♥t ❜❛♥❦s ♦❢❛ r✐✈❡r ✇✐t❤ ❝♦♥st❛♥t ✇✐❞t❤✳ ❲❡ ❝❛♥ ❛❝r♦ss t❤❡ r✐✈❡r ❜② ❛ ❜r✐❞❣❡ ✐♥ s✉❝❤ ❛✇❛② t❤❛t ✐t ✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ❜❛♥❦s✳ ❋✐♥❞ t❤❡ ❜❡st ♣♦s✐t✐♦♥ ❢♦r t❤❡ ❧❡❣s♦❢ t❤❡ ❜r✐❞❣❡ ❜② ♠✐♥✐♠✐③✐♥❣ t❤❡ s✉♠ ♦❢ ❞✐st❛♥❝❡s

AX +XY + Y B,

✇❤❡r❡ ❳ ❛♥❞ ❨ ❞❡♥♦t❡s t❤❡ ♣♦s✐t✐♦♥ ♦❢ t❤❡ ❧❡❣s ♦❢ t❤❡ ❜r✐❞❣❡✳

❋✐❣✉r❡ ✶✸✳✶✿ ❚❤❡ ♣r♦❜❧❡♠ ♦❢ t❤❡ ❜r✐❞❣❡

✶✺✶

✶✺✷ ❈❍❆P❚❊❘ ✶✸✳ ❈▲❆❙❙■❈❆▲ P❘❖❇▲❊▼❙ ■■

❋✐❣✉r❡ ✶✸✳✷✿ ❚❤❡ ♣r♦❜❧❡♠ ♦❢ t❤❡ ❝❛♠❡❧

❙♦❧✉t✐♦♥✳ ❙✐♥❝❡ t❤❡ r✐✈❡r ✐s ♦❢ ❝♦♥st❛♥t ✇✐❞t❤✱ t❤❡ ✐♥✈❛r✐❛♥t t❡r♠ ❳❨ ❝❛♥❜❡ ♦♠✐tt❡❞✳ ❚❤❡ tr❛♥s❧❛t✐♦♥ X 7→ X ′ = Y s❤♦✇s t❤❛t t❤❡ s✉❜ ✲ tr✐♣s ❆❳❛♥❞ ❨❇ ❝♦rr❡s♣♦♥❞ t♦ ❛ t✇♦ ✲ st❡♣s ❧♦♥❣ ♣♦❧②❣♦♥❛❧ ❝❤❛✐♥ ❢r♦♠ ❆✬ t♦ ❇✳ ❚❤❡str❛✐❣❤t ❧✐♥❡ s❡❣♠❡♥t ❆✬❇ ✐♥❞✐❝❛t❡s t❤❡ ♦♣t✐♠❛❧ ♣♦s✐t✐♦♥ ❢♦r t❤❡ ❧❡❣s ♦❢ t❤❡❜r✐❞❣❡✳

✶✸✳✷ ❚❤❡ ♣r♦❜❧❡♠ ♦❢ t❤❡ ❝❛♠❡❧

Pr♦❜❧❡♠✳ ❙✉♣♣♦s❡ t❤❛t t❤❡r❡ ❛r❡ t✇♦ ✈✐❧❧❛❣❡s ❆ ❛♥❞ ❇ ♦♥ t❤❡ s❛♠❡ ❜❛♥❦ ♦❢❛♥ ✉♥s✇❡r✈✐♥❣ r✐✈❡r✳ ❚❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡♠ ✐s t♦♦ ❧❛r❣❡ ❢♦r ❛ ❝❛♠❡❧ t♦✇❛❧❦ ❢r♦♠ ❆ t♦ ❇ ✇✐t❤♦✉t ❞r✐♥❦✐♥❣✳ ❋✐♥❞ t❤❡ ❜❡st ♣♦s✐t✐♦♥ ❢♦r t❤❡ ❝❛♠❡❧ t♦❤❛✈❡ ❛ ❞r✐♥❦ ❜② ♠✐♥✐♠✐③✐♥❣ t❤❡ s✉♠ ♦❢ ❞✐st❛♥❝❡s

AX +XB,

✇❤❡r❡ ❳ ❞❡♥♦t❡s t❤❡ ♣♦s✐t✐♦♥ ❛❧♦♥❣ t❤❡ r✐✈❡r✳❙♦❧✉t✐♦♥✳ ■♥st❡❛❞ ♦❢ ❛ r❡❞✉❝t✐♦♥ ❜② ❛♥ ✐♥✈❛r✐❛♥t q✉❛♥t✐t② ✭s❡❡ t❤❡ ♣r♦❜❧❡♠♦❢ t❤❡ ❜r✐❞❣❡✮ ✇❡ ✉s❡ ❛♥ ❡①♣❛♥s✐♦♥ ❜② ❛♥ ✐♥✈❛r✐❛♥t q✉❛♥t✐t② t♦ s♦❧✈❡ t❤❡♣r♦❜❧❡♠ ♦❢ t❤❡ ❝❛♠❡❧✿ ♠✐♥✐♠✐③❡ t❤❡ s✉♠

AA′ + A′X +XB,

✇❤❡r❡ ❆✬ ✐s t❤❡ ✐♠❛❣❡ ♦❢ ❆ ✉♥❞❡r t❤❡ r❡✢❡❝t✐♦♥ ❛❜♦✉t t❤❡ ❧✐♥❡ ♦❢ t❤❡ r✐✈❡r✳❚❤❡ str❛✐❣❤t ❧✐♥❡ s❡❣♠❡♥t ❆✬❇ ✐♥❞✐❝❛t❡s t❤❡ ♦♣t✐♠❛❧ ♣♦s✐t✐♦♥ ❢♦r t❤❡ ❝❛♠❡❧t♦ ❤❛✈❡ ❛ ❞r✐♥❦✳

✶✸✳✸ ❚❤❡ ❋❡r♠❛t ♣♦✐♥t ♦❢ ❛ tr✐❛♥❣❧❡

Pr♦❜❧❡♠✳ ❋✐♥❞ t❤❡ ♣♦✐♥t ♦❢ t❤❡ tr✐❛♥❣❧❡ ❆❇❈ ✇❤✐❝❤ ♠✐♥✐♠✐③❡s t❤❡ s✉♠ ♦❢❞✐st❛♥❝❡s

AX +BX + CX.

✶✸✳✸✳ ❚❍❊ ❋❊❘▼❆❚ P❖■◆❚ ❖❋ ❆ ❚❘■❆◆●▲❊ ✶✺✸

❋✐❣✉r❡ ✶✸✳✸✿ ❚❤❡ ❋❡r♠❛t ♣♦✐♥t ♦❢ ❛ tr✐❛♥❣❧❡

❙♦❧✉t✐♦♥✳ ❈♦♥s✐❞❡r ❛ r♦t❛t✐♦♥ ❛❜♦✉t ❇ ✇✐t❤ ❛♥❣❧❡ ✻✵ ❞❡❣r❡❡ ✐♥t♦ ❝❧♦❝❦✇✐s❡ ❞✐✲r❡❝t✐♦♥✳ ❙✐♥❝❡ t❤❡ tr✐❛♥❣❧❡ ❳❇❳✬ ✐s ❡q✉✐❧❛t❡r❛❧ t❤❡ s✉❜ ✲ tr✐♣ ❇❳ ❝❛♥ ❜❡ s✉❜st✐✲t✉t❡❞ ❜② ❳❳✬✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞ ❳❈❂❳✬❈✬ ❜❡❝❛✉s❡ r♦t❛t✐♦♥s ❛r❡ ✐s♦♠❡tr✐❡s✳❚❤❡r❡❢♦r❡ ❡✈❡r② ❝❤♦✐❝❡ ♦❢ ❳ ❝♦rr❡s♣♦♥❞s t♦ ❛ t❤r❡❡ ✲ st❡♣s ✲ ❧♦♥❣ ♣♦❧②❣♦♥❛❧❝❤❛✐♥ ❢r♦♠ ❆ t♦ ❈✬✳ ❙✐♥❝❡ t❤❡ str❛✐❣❤t ❧✐♥❡ s❡❣♠❡♥t ❆❈✬ ❣✐✈❡s t❤❡ ♠✐♥✐♠❛❧❧❡♥❣t❤ ✇❡ ❤❛✈❡ t❤❛t t❤❡ ♠✐♥✐♠✐③❡r s❛t✐s✜❡s t❤❡ ❝♦♥❞✐t✐♦♥s

6 AXB = 120◦

❛♥❞6 BXC = 6 BX ′C ′ = 120◦.

❋✐❣✉r❡ ✶✸✳✸ s❤♦✇s ❤♦✇ t♦ ❝♦♥str✉❝t t❤❡ ♠✐♥✐♠✐③❡r ❜② ✉s✐♥❣ ❡q✉✐❧❛t❡r❛❧ tr✐❛♥✲❣❧❡s ❧②✐♥❣ ♦♥ t❤❡ s✐❞❡s ♦❢ t❤❡ tr✐❛♥❣❧❡ ❆❇❈✳ ❚❤❡ ♠❡t❤♦❞ ❛♥❞ t❤❡ ❛r❣✉♠❡♥✲t❛t✐♦♥ ✐s ✇♦r❦✐♥❣ ❛s ❢❛r ❛s ❛❧❧ t❤❡ ❛♥❣❧❡s ♦❢ t❤❡ tr✐❛♥❣❧❡ ❆❇❈ ✐s ❧❡ss t❤❛♥ ✶✷✵❞❡❣r❡❡ ✐♥ ♠❡❛s✉r❡✳

❊①❝❡r❝✐s❡ ✶✸✳✸✳✶ ❊①♣❧❛✐♥ ✇❤② t❤❡ ♠❡t❤♦❞ ❢❛✐❧s ✐♥ ❝❛s❡ ♦❢ ❛♥ ❛♥❣❧❡ ♦❢ ♠❡❛✲s✉r❡ ❣r❡❛t❡r ♦r ❡q✉❛❧ t❤❛♥ ✶✷✵ ❞❡❣r❡❡✳

❘❡♠❛r❦ ■♥ ❝❛s❡ ♦❢ ❛ tr✐❛♥❣❧❡ ❤❛✈✐♥❣ ❛♥ ❛♥❣❧❡ ♦❢ ♠❡❛s✉r❡ ❣r❡❛t❡r ♦r ❡q✉❛❧t❤❛♥ ✶✷✵ ❞❡❣r❡❡ t❤❡ s♦❧✉t✐♦♥ ✐s ❥✉st t❤❡ ✈❡rt❡① ✇❤❡r❡ t❤❡ ❝r✐t✐❝❛❧ ✈❛❧✉❡ ♦❢ t❤❡♠❡❛s✉r❡ ✐s ❛tt❛✐♥❡❞ ♦r ❡①❝❡❡❞❡❞✳

❉❡✜♥✐t✐♦♥ ❚❤❡ ♣♦✐♥t ♦❢ t❤❡ tr✐❛♥❣❧❡ ❆❇❈ ✇❤✐❝❤ ♠✐♥✐♠✐③❡s t❤❡ s✉♠ ♦❢ ❞✐s✲t❛♥❝❡s

AX +BX + CX

✐s ❝❛❧❧❡❞ t❤❡ ❋❡r♠❛t✲♣♦✐♥t ♦❢ t❤❡ tr✐❛♥❣❧❡✳

✶✺✹ ❈❍❆P❚❊❘ ✶✸✳ ❈▲❆❙❙■❈❆▲ P❘❖❇▲❊▼❙ ■■

❈❤❛♣t❡r ✶✹

▲♦♥❣✐t✉❞❡s ❛♥❞ ❧❛t✐t✉❞❡s

Pr♦❜❧❡♠✳ ❋✐♥❞ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ❆ ❛♥❞ ❇ ♦♥ t❤❡ s✉r❢❛❝❡ ♦❢ t❤❡ ❊❛rt❤✳❙♦❧✉t✐♦♥✳ ■♥ ❣❡♦❣r❛♣❤② t❤❡ ❧♦♥❣✐t✉❞❡ ❛♥❞ t❤❡ ❧❛t✐t✉❞❡ ❛r❡ ✉s❡❞ t♦ ❞❡t❡r♠✐♥❡♣♦s✐t✐♦♥s ♦♥ t❤❡ s✉r❢❛❝❡ ♦❢ t❤❡ ❊❛rt❤✳ ❚❤❡ ❧♦♥❣✐t✉❞❡ λ ✐s ❛ r♦t❛t✐♦♥❛❧ ❛♥❣❧❡ t♦s♣❡❝✐❢② t❤❡ ❡❛st✲✇❡st ♣♦s✐t✐♦♥ ♦❢ t❤❡ ♣♦✐♥t r❡❧❛t✐✈❡ t♦ t❤❡ ●r❡❡♥✇✐❝❤ ♠❡r✐❞✐❛♥❛❝r♦ss ❘♦②❛❧ ❖❜s❡r✈❛t♦r②✱ ●r❡❡♥✇✐❝❤✳ ❚❤❡ ❧❛t✐t✉❞❡ ϕ ✐s t❤❡ ✐♥❝❧✐♥❛t✐♦♥ ❛♥❣❧❡r❡❧❛t✐✈❡ t♦ t❤❡ ♣❧❛♥❡ ♦❢ t❤❡ ❊q✉❛t♦r✳ ■♥ ✇❤❛t ❢♦❧❧♦✇s ✇❡ ✇✐❧❧ ✉s❡ t❤❡ s✐❣♥s ✰❛♥❞ ✲ ✐♥st❡❛❞ ♦❢ ♥♦rt❤ ❛♥❞ s♦✉t❤ ♦r ❡❛st ❛♥❞ ✇❡st✳ ❚♦ s✐♠♣❧✐❢② t❤❡ ❢♦r♠✉❧❛s✐♥ t❤❡ ❝❛❧❝✉❧❛t✐♦♥ ✇❡ s✉♣♣♦s❡ t❤❛t t❤❡ r❛❞✐✉s ♦❢ t❤❡ ❊❛rt❤ ✐s ✶ ✉♥✐t✳❋✐rst st❡♣ P②t❤❛❣♦r❡❛♥ t❤❡♦r❡♠ ✐♥ t❤❡ r✐❣❤t tr✐❛♥❣❧❡ ❆❇❈ ❣✐✈❡s t❤❡ ❊✉✲❝❧✐❞❡❛♥ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ❆ ❛♥❞ ❇✿

AB2 = AC2 + CB2.

❙✐♥❝❡ ❆❈ ✐s t❤❡ ✈❡rt✐❝❛❧ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ♣♦✐♥ts✱

AC2 = (AA′ − BB′)2 = (sinϕA − sinϕB)2.

❋✐❣✉r❡ ✶✹✳✶✿ ▲♦♥❣✐t✉❞❡s ❛♥❞ ❧❛t✐t✉❞❡s ■

✶✺✺

✶✺✻ ❈❍❆P❚❊❘ ✶✹✳ ▲❖◆●■❚❯❉❊❙ ❆◆❉ ▲❆❚■❚❯❉❊❙

❋✐❣✉r❡ ✶✹✳✷✿ ▲♦♥❣✐t✉❞❡s ❛♥❞ t❤❡ ❧❛t✐t✉❞❡s ■■

❚♦ ❝♦♠♣✉t❡ ❈❇ ❝♦♥s✐❞❡r t❤❡ ♣r♦❥❡❝t❡❞ s❡❣♠❡♥t ❆✬❇✬ ✐♥ t❤❡ ❡q✉❛t♦r✐❛❧ ♣❧❛♥❡✿

OA′ = cosϕA, OB′ = cosϕB ❛♥❞ 6 A′OB′ = λB − λA.

❯s✐♥❣ t❤❡ ❝♦s✐♥❡ r✉❧❡ ✇❡ ❤❛✈❡ t❤❛t

A′B′2 = cos2 ϕA + cos2 ϕB − 2 cosϕA cosϕB cos(λB − λA) =

cos2 ϕA + cos2 ϕB − 2 cosϕA cosϕB(cosλB cosλA + sinλB sinλA) =(

cosϕA cosλA−cosϕB cosλB

)2

+

(

cosϕA sinλA−cosϕB sinλB

)2

= CB2.

❚❤❡r❡❢♦r❡

AB2 = (cosϕA cosλA − cosϕB cosλB)2 + (cosϕA sinλA − cosϕB sinλB)

2+

(sinϕA − sinϕB)2.

❙❡❝♦♥❞ st❡♣✳ ❯s✐♥❣ ❆❇ ✇❡ ❝❛♥ ❝♦♠♣✉t❡ t❤❡ ❝❡♥tr❛❧ ❛♥❣❧❡ ω ✐♥ t❤❡ tr✐❛♥❣❧❡❆❖❇ ❜② t❤❡ ❝♦s✐♥❡ r✉❧❡

AB2 = 2− 2 cosω

❜❡❝❛✉s❡ t❤❡ r❛❞✐✉s ♦❢ t❤❡ ❊❛rt❤ ✐s ❝❤♦s❡♥ ❛s ❛ ✉♥✐t✳❚❤✐r❞ st❡♣✳ ❚❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ❆ ❛♥❞ ❇ ♦♥ t❤❡ s✉r❢❛❝❡ ♦❢ t❤❡ ❊❛rt❤ ✐s❥✉st t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ s❤♦rt❡r ❛r❝ ❥♦✐♥✐♥❣ ❆ ❛♥❞ ❇ ❛❧♦♥❣ t❤❡ ❝✐r❝❧❡ ❝✉tt❡❞ ❜②t❤❡ ♣❧❛♥❡ ❆❖❇✿

t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❛r❝ ❢r♦♠ ❆ t♦ ❇ = ω ✭✐♥ r❛❞✐❛♥✮.

❘❡♠❛r❦ ❚❤❡ ❞✐st❛♥❝❡ ✐♥ ❦✐❧♦♠❡t❡rs ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❢r♦♠ t❤❡ ❢♦r♠✉❧❛

t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❛r❝ ❢r♦♠ ❆ t♦ ❇R

= ω,

✇❤❡r❡ R ≈ 6378.1 ❦♠✳

✶✺✼

❊①❝❡r❝✐s❡ ✶✹✳✵✳✷ ❋✐♥❞ t❤❡ ❞✐st❛♥❝❡s ❜❡t✇❡❡♥ ❲♦r❧❞ ❝✐t✐❡s ♦♥ t❤❡ s✉r❢❛❝❡ ♦❢t❤❡ ❊❛rt❤✳

❈✐t② ▲❛t✐t✉❞❡ ▲♦♥❣✐t✉❞❡❆❜❡r❞❡❡♥✱ ❙❝♦t❧❛♥❞ ✺✼ ◆ ✷ ❲❇✉❞❛♣❡st✱ ❍✉♥❣❛r② ✹✼ ◆ ✶✾ ❊

❈❛✐r♦✱ ❊❣②♣t ✸✵ ◆ ✸✶ ❊❉❛❦❛r✱ ❙❡♥❡❣❛❧ ✶✹ ◆ ✶✼ ❲

❊❞✐♥❜✉r❣❤✱ ❙❝♦t❧❛♥❞ ✺✺ ◆ ✸ ❲❋r❛♥❦❢✉rt✱ ●❡r♠❛♥② ✺✵ ◆ ✽ ❊●❡♦r❣❡t♦✇♥✱ ●✉②❛♥❛ ✻ ◆ ✺✽ ❲❍❛♠❜✉r❣✱ ●❡r♠❛♥② ✺✸ ◆ ✶✵ ❊■r❦✉ts❦✱ ❘✉ss✐❛ ✺✷ ◆ ✶✵✹ ❊

❏❛❦❛rt❛✱ ■♥❞♦♥❡s✐❛ ✻ ❙ ✶✵✻ ❊❑✐♥❣st♦♥❡✱ ❏❛♠❛✐❝❛ ✶✼ ◆ ✼✻ ❲▲❛ P❛③✱ ❇♦❧✐✈✐❛ ✶✻ ❙ ✻✽ ❲▼❛❞r✐❞✱ ❙♣❛✐♥ ✹✵ ◆ ✸ ❲◆❛❣❛s❛❦✐✱ ❏❛♣❛♥ ✸✷ ◆ ◆ ✶✷✷ ❊❖❞❡ss❛✱ ❯❦r❛✐♥❡ ✹✻ ◆ ✸✵ ❊P❛r✐s✱ ❋r❛♥❝❡ ✹✽ ◆ ✷✵ ❊

❘✐♦ ❞❡ ❏❛♥❡✐r♦✱ ❇r❛s✐❧ ✷✷ ❙ ✹✸ ❲❙②❞♥❡②✱ ❆✉str❛❧✐❛ ✸✹ ❙ ✶✺✶ ❊

❚❛♥❛♥❛r✐✈❡✱ ▼❛❞❛❣❛s❝❛r ✶✽ ❙ ✹✼ ❊❱❡r❛❝r✉③✱ ▼❡①✐❝♦ ✶✾ ◆ ✾✻ ❲❲❛rs❛✇✱ P♦❧❛♥❞ ✺✷ ◆ ✷✶ ❊

❩ür✐❝❤✱ ❙✇✐t③❡r❧❛♥❞ ✹✼ ◆ ✽ ❊

✶✺✽ ❈❍❆P❚❊❘ ✶✹✳ ▲❖◆●■❚❯❉❊❙ ❆◆❉ ▲❆❚■❚❯❉❊❙

P❛rt ■■

❆♥❛❧②t✐❝❛❧ ●❡♦♠❡tr②

✶✺✾

❈❤❛♣t❡r ✶✺

❘❡❝t❛♥❣✉❧❛r ❈❛rt❡s✐❛♥

❈♦♦r❞✐♥❛t❡s ✐♥ ❛ P❧❛♥❡

✶✺✳✶ ❈♦♦r❞✐♥❛t❡s ✐♥ ❛ ♣❧❛♥❡

▲❡t ✉s ❞r❛✇ ✐♥ t❤❡ ♣❧❛♥❡ t✇♦ ♠✉t✉❛❧❧② ♣❡r♣❡♥❞✐❝✉❧❛r ✐♥t❡rs❡❝t✐♥❣ ❧✐♥❡sOx ❛♥❞Oy ✇❤✐❝❤ ❛r❡ t❡r♠❡❞ ❝♦♦r❞✐♥❛t❡ ❛①❡s ✭❋✐❣✳ ✶✺✳✶✮✳ ❚❤❡ ♣♦✐♥t ♦❢ ✐♥t❡rs❡❝t✐♦♥O ♦❢ t❤❡ t✇♦ ❛①❡s ✐s ❝❛❧❧❡❞ t❤❡ ♦r✐❣✐♥ ♦❢ ❝♦♦r❞✐♥❛t❡s✱ ♦r s✐♠♣❧② t❤❡ ♦r✐❣✐♥✳ ■t❞✐✈✐❞❡s ❡❛❝❤ ♦❢ t❤❡ ❛①❡s ✐♥t♦ t✇♦ s❡♠✐✲❛①❡s✳ ❖♥❡ ♦❢ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ s❡♠✐✲❛①❡s ✐s ❝♦♥✈❡♥t✐♦♥❛❧❧② ❝❛❧❧❡❞ ♣♦s✐t✐✈❡ ✭✐♥❞✐❝❛t❡❞ ❜② ❛♥ ❛rr♦✇ ✐♥ t❤❡ ❞r❛✇✐♥❣✮✱t❤❡ ♦t❤❡r ❜❡✐♥❣ ♥❡❣❛t✐✈❡✳

❋✐❣✉r❡ ✶✺✳✶✿ ❚❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠

❆♥② ♣♦✐♥t A ✐♥ ❛ ♣❧❛♥❡ ✐s s♣❡❝✐✜❡❞ ❜② ❛ ♣❛✐r ♦❢ ♥✉♠❜❡rs ✕ ❝❛❧❧❡❞ t❤❡r❡❝t❛♥❣✉❧❛r ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ♣♦✐♥t A ✕ t❤❡ ❛❜s❝✐ss❛ (x) ❛♥❞ t❤❡ ♦r❞✐♥❛t❡(y) ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ r✉❧❡✳

✶✻✶

✶✻✷ ❈❍❆P❚❊❘ ✶✺✳ ❈❆❘❚❊❙■❆◆ ❈❖❖❘❉■◆❆❚❊❙ ■◆ ❆ P▲❆◆❊

❋✐❣✉r❡ ✶✺✳✷✿ ❈♦♦r❞✐♥❛t❡s ♦❢ ❛ ♣♦✐♥t

❚❤r♦✉❣❤ t❤❡ ♣♦✐♥t A ✇❡ ❞r❛✇ ❛ str❛✐❣❤t ❧✐♥❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ ❛①✐s ♦❢♦r❞✐♥❛t❡s (Oy) t♦ ✐♥t❡rs❡❝t t❤❡ ❛①✐s ♦❢ ❛❜s❝✐ss❛s (Ox) ❛t s♦♠❡ ♣♦✐♥t Ax

✭❋✐❣✳ ✶✺✳✷✮✳ ❚❤❡ ❛❜s❝✐ss❛ ♦❢ t❤❡ ♣♦✐♥t A s❤♦✉❧❞ ❜❡ ✉♥❞❡rst♦♦❞ ❛s ❛ ♥✉♠❜❡rx ✇❤♦s❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡ ✐s ❡q✉❛❧ t♦ t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ O t♦ Ax ✇❤✐❝❤ ✐s ♣♦s✐✲t✐✈❡ ✐❢ Ax ❜❡❧♦♥❣ t♦ t❤❡ ♣♦s✐t✐✈❡ s❡♠✐✲❛①✐s ❛♥❞ ♥❡❣❛t✐✈❡ ✐❢ Ax ❜❡❧♦♥❣s t♦ t❤❡♥❡❣❛t✐✈❡ s❡♠✐✲❛①✐s✳ ■❢ t❤❡ ♣♦✐♥t Ax ❝♦✐♥❝✐❞❡s ✇✐t❤ t❤❡ ♦r✐❣✐♥✱ t❤❡♥ ✇❡ ♣✉t x❡q✉❛❧ t♦ ③❡r♦✳

❚❤❡ ♦r❞✐♥❛t❡ (y) ♦❢ t❤❡ ♣♦✐♥t A ✐s ❞❡t❡r♠✐♥❡❞ ✐♥ ❛ s✐♠✐❧❛r ✇❛②✳❲❡ s❤❛❧❧ ✉s❡ ❢♦❧❧♦✇✐♥❣ ♥♦t❛t✐♦♥✿ A(x, y) ✇❤✐❝❤ ♠❡❛♥s t❤❛t t❤❡ ❝♦♦r❞✐♥❛t❡s

♦❢ t❤❡ ♣♦✐♥t A ❛r❡ x ✭❛❜s❝✐ss❛✮ ❛♥❞ (y) ✭♦r❞✐♥❛t❡✮✳❚❤❡ ❝♦♦r❞✐♥❛t❡ ❛①❡s s❡♣❛r❛t❡ t❤❡ ♣❧❛♥❡ ✐♥t♦ ❢♦✉r r✐❣❤t ❛♥❣❧❡s t❡r♠❡❞

t❤❡ q✉❛❞r❛♥ts ❛s s❤♦✇♥ ✐♥ ❋✐❣✳ ✶✺✳✸✳ ❲✐t❤✐♥ t❤❡ ❧✐♠✐ts ♦❢ ♦♥❡ q✉❛❞r❛♥tt❤❡ s✐❣♥s ♦❢ ❜♦t❤ ❝♦♦r❞✐♥❛t❡s r❡♠❛✐♥ ✉♥❝❤❛♥❣❡❞✳ ❆s ✇❡ s❡❡ ✐♥ t❤❡ ✜❣✉r❡✱t❤❡ q✉❛❞r❛♥ts ❛r❡ ❞❡♥♦t❡❞ ❛♥❞ ❝❛❧❧❡❞ t❤❡ ✜rst✱ s❡❝♦♥❞✱ t❤✐r❞✱ ❛♥❞ ❢♦✉rt❤ ❛s❝♦✉♥t❡❞ ❛♥t✐❝❧♦❝❦✇✐s❡ ❜❡❣✐♥♥✐♥❣ ✇✐t❤ t❤❡ q✉❛❞r❛♥t ✐♥ ✇❤✐❝❤ ❜♦t❤ ❝♦♦r❞✐♥❛t❡s❛r❡ ♣♦s✐t✐✈❡✳

■❢ ❛ ♣♦✐♥t ❧✐❡s ♦♥ t❤❡ x✲❛①✐s ✭✐✳❡✳ ♦♥ t❤❡ ❛①✐s ♦❢ ❛❜s❝✐ss❛s✮ t❤❡♥ ✐ts ♦r❞✐♥❛t❡y ✐s ❡q✉❛❧ t♦ ③❡r♦❀ ✐❢ ❛ ♣♦✐♥t ❧✐❡s ♦♥ t❤❡ y✲❛①✐s✱ ✭✐✳❡✳ ♦♥ t❤❡ ❛①✐s ♦❢ ♦r❞✐♥❛t❡s✮✱t❤❡♥ ✐ts ❛❜s❝✐ss❛ x ✐s ③❡r♦✳ ❚❤❡ ❛❜s❝✐ss❛ ❛♥❞ ♦r❞✐♥❛t❡ ♦❢ t❤❡ ♦r✐❣✐♥ ✭✐✳❡✳ ♦❢t❤❡ ♣♦✐♥t O✮ ❛r❡ ❡q✉❛❧ ③❡r♦✳

❚❤❡ ♣❧❛♥❡ ♦♥ ✇❤✐❝❤ t❤❡ ❝♦♦r❞✐♥❛t❡s x ❛♥❞ y ❛r❡ ✐♥tr♦❞✉❝❡❞ ❜② t❤❡ ❛❜♦✈❡♠❡t❤♦❞ ✇✐❧❧ ❜❡ ❝❛❧❧❡❞ t❤❡ xy✲♣❧❛♥❡✳ ❆♥ ❛r❜✐tr❛r② ♣♦✐♥t ✐♥ t❤✐s ✇✐t❤ t❤❡❝♦♦r❞✐♥❛t❡s x ❛♥❞ y ✇✐❧❧ s♦♠❡t✐♠❡s ❜❡ ❞❡♥♦t❡❞ s✐♠♣❧② (x, y)✳

❋♦r ❛♥ ❛r❜✐tr❛r② ♣❛✐r ♦❢ r❡❛❧ ♥✉♠❜❡rs x ❛♥❞ y t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ♣♦✐♥tA ✐♥ t❤❡ xy✲♣❧❛♥❡ ❢♦r ✇❤✐❝❤ x ✇✐❧❧ ❜❡ ✐ts ❛❜s❝✐ss❛ ❛♥❞ y ✐ts ♦r❞✐♥❛t❡✳

■♥❞❡❡❞✱ s✉♣♣♦s❡ ❢♦r ❞❡✜♥✐t❡♥❡ss x > 0✱ ❛♥❞ y < 0✳ ▲❡t ✉s t❛❦❡ ♦♥ t❤❡

✶✺✳✶✳ ❈❖❖❘❉■◆❆❚❊❙ ■◆ ❆ P▲❆◆❊ ✶✻✸

❋✐❣✉r❡ ✶✺✳✸✿ ❈♦♦r❞✐♥❛t❡s ♦❢ ❛ ♣♦✐♥t

♣♦s✐t✐✈❡ s❡♠✐✲❛①✐s x ❛ ♣♦✐♥t Ax ❛t t❤❡ ❞✐st❛♥❝❡ x ❢r♦♠ t❤❡ ♦r✐❣✐♥ O✱ ❛♥❞ ❛♣♦✐♥t Ay ♦♥ t❤❡ ♥❡❣❛t✐✈❡ s❡♠✐✲❛①✐s y ❛t t❤❡ ❞✐st❛♥❝❡ |y| ❢r♦♠ O✳ ❲❡ t❤❡♥❞r❛✇ t❤r♦✉❣❤ t❤❡ ♣♦✐♥ts Ax ❛♥❞ Ay str❛✐❣❤t ❧✐♥❡s ♣❛r❛❧❧❡❧ t♦ t❤❡ ❛①❡s y ❛♥❞x✱ r❡s♣❡❝t✐✈❡❧② ✭❋✐❣✳ ✶✺✳✹✮✳ ❚❤❡s❡ ❧✐♥❡s ✇✐❧❧ ✐♥t❡rs❡❝t ❛t ❛ ♣♦✐♥t A ✇❤♦s❡❛❜s❝✐ss❛ ✐s ♦❜✈✐♦✉s❧② x✱ ❛♥❞ ♦r❞✐♥❛t❡ ✐s y✳ ■♥ ♦t❤❡r ❝❛s❡ ✭x < 0✱ y > 0❀ x > 0✱y > 0 ❛♥❞ x < 0✱ y < 0✮ t❤❡ ♣r♦♦❢ ✐s ❛♥❛❧♦❣♦✉s✳

❋✐❣✉r❡ ✶✺✳✹✿ ❊①❛♠♣❧❡ ♦❢ ❝♦♦r❞✐♥❛t❡s

▲❡t ✉s ❝♦♥s✐❞❡r s❡✈❡r❛❧ ✐♠♣♦rt❛♥t ❝❛s❡s ♦❢ ❛♥❛❧②t✐❝❛❧ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢❞♦♠❛✐♥s ♦♥ t❤❡ xy✲♣❧❛♥❡ ✇✐t❤ t❤❡ ❛✐❞ ♦❢ ✐♥❡q✉❛❧✐t✐❡s✳ ❆ s❡t ♦❢ ♣♦✐♥ts ♦❢ t❤❡xy✲♣❧❛♥❡ ❢♦r ✇❤✐❝❤ x > a ✐s ❛ ❤❛❧❢✲♣❧❛♥❡ ❜♦✉♥❞❡❞ ❜② ❛ str❛✐❣❤t ❧✐♥❡ ♣❛ss✐♥❣t❤r♦✉❣❤ t❤❡ ♣♦✐♥t (a, 0) ♣❛r❛❧❧❡❧ t♦ t❤❡ ❛①✐s ♦❢ ♦r❞✐♥❛t❡s ✭❋✐❣✳ ✶✺✳✺✮✳ ❆ s❡t♦❢ ♣♦✐♥ts ❢♦r ✇❤✐❝❤ a < x < b r❡♣r❡s❡♥ts t❤❡ ✐♥t❡rs❡❝t✐♦♥ ✭✐✳❡✳ t❤❡ ❝♦♠♠♦♥♣♦rt✐♦♥✮ ♦❢ t❤❡ ❤❛❧❢✲♣❧❛♥❡s s♣❡❝✐✜❡❞ ❜② t❤❡ ✐♥❡q✉❛❧✐t✐❡s a < x ❛♥❞ x < b✳❚❤✉s✱ t❤✐s s❡t ✐s ❛ ❜❛♥❞ ❜❡t✇❡❡♥ t❤❡ str❛✐❣❤t ❧✐♥❡s ♣❛r❛❧❧❡❧ t♦ t❤❡ y✲❛①✐s ❛♥❞

✶✻✹ ❈❍❆P❚❊❘ ✶✺✳ ❈❆❘❚❊❙■❆◆ ❈❖❖❘❉■◆❆❚❊❙ ■◆ ❆ P▲❆◆❊

❋✐❣✉r❡ ✶✺✳✺✿ ❊①❛♠♣❧❡ ♦❢ ❛ ❤❛❧❢ ♣❧❛♥❡ ❛♥❞ ❛ str✐♣

❋✐❣✉r❡ ✶✺✳✻✿ ❊①❛♠♣❧❡ ♦❢ ❛ r❡❝t❛♥❣❧❡

♣❛ss✐♥❣ t❤r♦✉❣❤ t❤❡ ♣♦✐♥ts (a, 0) ❛♥❞ (b, 0) ✭❋✐❣✳ ✶✺✳✺✮✳ ❆ s❡t ♦❢ ♣♦✐♥ts ❢♦r✇❤✐❝❤ a < x < b✱ c < y < d ✐s ❛ r❡❝t❛♥❣❧❡ ✇✐t❤ ✈❡rt✐❝❡s ❛t ♣♦✐♥ts ❢♦r ✇❤✐❝❤a < x < b✱ c < y < d ✐s ❛ r❡❝t❛♥❣❧❡ ✇✐t❤ ✈❡rt✐❝❡s ❛t ♣♦✐♥ts (a, c) (a, d)✱ (b, c)✱(b, d)✳ ✭❋✐❣✳ ✶✺✳✻✮

■♥ ❝♦♥❝❧✉s✐♦♥✱ ❧❡t ✉s s♦❧✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦❜❧❡♠✿ ❋✐♥❞ t❤❡ ❛r❡❛ ♦❢ ❛tr✐❛♥❣❧❡ ✇✐t❤ ✈❡rt✐❝❡s ❛t ♣♦✐♥ts A1(x1, y1)✱ A2(x2, y2)✱ A3(x3, y3)✳ ▲❡t t❤❡ tr✐✲❛♥❣❧❡ ❜❡ ❧♦❝❛t❡❞ r❡❧❛t✐✈❡ t♦ t❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ❛s ✐s s❤♦✇♥ ✐♥ ❋✐❣✳ ✶✺✳✼✳■♥ t❤✐s ♣♦s✐t✐♦♥ ✐ts ❛r❡❛ ✐s ❡q✉❛❧ t♦ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❛r❡❛ tr❛♣❡③✲✐✉♠ B1A1A3B3 ❛♥❞ t❤❡ s✉♠ ♦❢ t❤❡ ❛r❡❛s ♦❢ t❤❡ tr❛♣❡③✐❛ B1A1A2B2 ❛♥❞B2A2A3B3✳

❚❤❡ ❜❛s❡s ♦❢ t❤❡ tr❛♣❡③✐✉♠ B1A1A3B3 ❛r❡ ❡q✉❛❧ t♦ y1 ❛♥❞ y3✱ ✐ts ❛❧t✐t✉❞❡❜❡✐♥❣ ❡q✉❛❧ t♦ x3 − x1✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❛r❡❛ ♦❢ t❤❡ tr❛♣❡③✐✉♠

S(B1A1A3B3) =1

2(y3 + y1)(x3 − x1).

✶✺✳✷✳ ❊❳❊❘❈■❙❊❙ ✶✻✺

❋✐❣✉r❡ ✶✺✳✼✿ ❆r❡❛ ♦❢ ❛ tr✐❛♥❣❧❡

❚❤❡ ❛r❡❛s ♦❢ t✇♦ ♦t❤❡r tr❛♣❡③✐❛ ❛r❡ ❢♦✉♥❞ ❛♥❛❧♦❣♦✉s❧②✿

S(B1A1A2B2) =1

2(y2 + y1)(x2 − x1),

S(B2A2A3B3) =1

2(y3 + y2)(x3 − x2).

❚❤❡ ❛r❡❛ ♦❢ t❤❡ tr✐❛♥❣❧❡ A1A2A3✿

S(A1A2A3) =1

2(y3 + y1)(x3 − x1)

− 1

2(y2 + y1)(x2 − x1)−

1

2(y3 + y2)(x3 − x2)

=1

2(x2y3 − y3x1 + x1y2 − y2x3 + x3y1 − y1x2).

❚❤✐s ❢♦r♠✉❧❛ ❝❛♥ ❜❡ r❡✇r✐tt❡♥ ✐♥ ❛ ❝♦♥✈❡♥✐❡♥t ❢♦r♠✿

S(A1A2A3) =1

2{(y3 − y1)(x2 − x1)− (y2 − y1)(x3 − x1)}.

❚❤♦✉❣❤ t❤❡ ❛❜♦✈❡ ❢♦r♠✉❧❛ ❢♦r ❝♦♠♣✉t✐♥❣ t❤❡ ❛r❡❛ ♦❢ t❤❡ tr✐❛♥❣❧❡ ❤❛s❜❡❡♥ ❞❡r✐✈❡❞ ❢♦r ❛ ♣❛rt✐❝✉❧❛r ❧♦❝❛t✐♦♥ ♦❢ t❤❡ tr✐❛♥❣❧❡ r❡❧❛t✐✈❡ t♦ t❤❡ ❝♦♦r❞✐♥❛t❡s②st❡♠✱ ✐t ②✐❡❧❞s ❛ ❝♦rr❡❝t r❡s✉❧t ✭✉♣ t♦ ❛ s✐❣♥✮ ❢♦r ❛♥② ♣♦s✐t✐♦♥ ♦❢ t❤❡ tr✐❛♥❣❧❡✳❚❤✐s ✇✐❧❧ ❜❡ ♣r♦✈❡❞ ❧❛t❡r ♦♥ ✭✐♥ ❙❡❝t✐♦♥ ❳❳❳❳✮✳

✶✺✳✷ ❊①❡r❝✐s❡s

✶✳ ❲❤❛t ✐s t❤❡ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ ♣♦✐♥ts ♦❢ t❤❡ xy✲♣❧❛♥❡ ❢♦r ✇❤✐❝❤ ✭❛✮ |x| = a✱✭❜✮ |x| = |y|❄

✶✻✻ ❈❍❆P❚❊❘ ✶✺✳ ❈❆❘❚❊❙■❆◆ ❈❖❖❘❉■◆❆❚❊❙ ■◆ ❆ P▲❆◆❊

✷✳ ❲❤❛t ✐s t❤❡ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ ♣♦✐♥ts ♦❢ t❤❡ xy✲♣❧❛♥❡ ❢♦r ✇❤✐❝❤ ✭❛✮ |x| < a✱✭❜✮ |x| < a✱ |y| < b❄

✸✳ ❋✐♥❞ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ❛ ♣♦✐♥t s②♠♠❡tr✐❝❛❧ t♦ t❤❡ ♣♦✐♥t A(x, y) ❛❜♦✉tt❤❡ x✲❛①✐s ✭y✲❛①✐s✱ t❤❡ ♦r✐❣✐♥✮✳

✹✳ ❋✐♥❞ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ❛ ♣♦✐♥t s②♠♠❡tr✐❝❛❧ t♦ t❤❡ ♣♦✐♥t A(x, y) ❛❜♦✉tt❤❡ ❜✐s❡❝t♦r ♦❢ t❤❡ ✜rst ✭s❡❝♦♥❞✮ q✉❛❞r❛♥t✳

✺✳ ❍♦✇ ✇✐❧❧ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ♣♦✐♥t A(x, y) ❝❤❛♥❣❡ ✐❢ t❤❡ y✲❛①✐s ✐st❛❦❡♥ ❢♦r t❤❡ x✲❛①✐s✱ ❛♥❞ ✈✐❝❡ ✈❡rs❛❄

✻✳ ❍♦✇ ✇✐❧❧ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ♣♦✐♥t A(x, y) ❝❤❛♥❣❡ ✐❢ t❤❡ ♦r✐❣✐♥ ✐s❞✐s♣❧❛❝❡❞ ✐♥t♦ t❤❡ ♣♦✐♥t A0(x0, y0) ✇✐t❤♦✉t ❝❤❛♥❣✐♥❣ t❤❡ ❞✐r❡❝t✐♦♥s ♦❢ t❤❡❝♦♦r❞✐♥❛t❡ ❛①❡s❄

✼✳ ❋✐♥❞ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ♠✐❞✲♣♦✐♥ts ♦❢ t❤❡ s✐❞❡s ♦❢ ❛ sq✉❛r❡ t❛❦✐♥❣✐ts ❞✐❛❣♦♥❛❧s ❢♦r t❤❡ ❝♦♦r❞✐♥❛t❡ ❛①❡s✳

✽✳ ■t ✐s ❦♥♦✇♥ t❤❛t t❤r❡❡ ♣♦✐♥ts (x1, y1)✱ (x2, y2)✱ (x3, y3) ❛r❡ ❝♦❧❧✐♥❡❛r✳❍♦✇ ❝❛♥ ♦♥❡ ✜♥❞ ♦✉t ✇❤✐❝❤ ♦❢ t❤❡s❡ ♣♦✐♥ts ✐s s✐t✉❛t❡❞ ❜❡t✇❡❡♥ t❤❡ ♦t❤❡rt✇♦❄

✶✺✳✸ ❚❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ♣♦✐♥ts

▲❡t t❤❡r❡ ❜❡ ❣✐✈❡♥ ♦♥ t❤❡ xy✲♣❧❛♥❡ t✇♦ ♣♦✐♥ts✿ A1 ✇✐t❤ t❤❡ ❝♦♦r❞✐♥❛t❡s x1✱y1 ❛♥❞ A2 ✇✐t❤ t❤❡ ❝♦♦r❞✐♥❛t❡s x2✱ y2✳ ■t ✐s r❡q✉✐r❡❞ t♦ ❡①♣r❡ss t❤❡ ❞✐st❛♥❝❡❜❡t✇❡❡♥ t❤❡ ♣♦✐♥ts A1 ❛♥❞ A2 ✐♥ t❡r♠s ♦❢ t❤❡✐r ❝♦♦r❞✐♥❛t❡s✳

❙✉♣♣♦s❡ x1 6= x2 ❛♥❞ y1 6= y2✳ ❚❤r♦✉❣❤ t❤❡ ♣♦✐♥ts A1 ❛♥❞ A2 ✇❡ ❞r❛✇str❛✐❣❤t ❧✐♥❡s ♣❛r❛❧❧❡❧ t♦ t❤❡ ❝♦♦r❞✐♥❛t❡ ❛①❡s ✭❋✐❣✳ ✶✺✳✽✮✳ ❚❤❡ ❞✐st❛♥❝❡ ❜❡✲t✇❡❡♥ t❤❡ ♣♦✐♥ts A ❛♥❞ A1 ✐s ❡q✉❛❧ t♦ |y1−y2|✱ ❛♥❞ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡♣♦✐♥ts A ❛♥❞ A2 ✐s ❡q✉❛❧ t♦ |x1 − x2|✳ ❆♣♣❧②✐♥❣ t❤❡ P②t❤❛❣♦r❡❛♥ t❤❡♦r❡♠ t♦t❤❡ r✐❣❤t✲❛♥❣❧❡❞ tr✐❛♥❣❧❡ A1AA2✱ ✇❡ ❣❡t

(x1 − x2)2 + (y1 − y2)

2 = d2, (∗)

❚❤♦✉❣❤ t❤❡ ❢♦r♠✉❧❛ (∗) ❢♦r ❞❡t❡r♠✐♥✐♥❣ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ♣♦✐♥ts ❤❛s❜❡❡♥ ❞❡r✐✈❡❞ ❜② ✉s ♣r♦❝❡❡❞✐♥❣ ❢r♦♠ t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t x1 6= x2✱ y1 6= y2✱ ✐tr❡♠❛✐♥s tr✉❡ ❢♦r ♦t❤❡r ❝❛s❡s ❛s ✇❡❧❧✳ ■♥❞❡❡❞✱ ❢♦r x1 = x2✱ y1 6= y2 d ✐s ❡q✉❛❧t♦ |y1 − y2| ✭❋✐❣✳ ✶✺✳✾✮✳ ❚❤❡ s❛♠❡ r❡s✉❧t ✐s ♦❜t❛✐♥❡❞ ✉s✐♥❣ t❤❡ ❢♦r♠✉❧❛ (∗)✳❋♦r x1 6= x2✱ y1 = y2 ✇❡ ❣❡t ❛ s✐♠✐❧❛r r❡s✉❧t✳ ■❢ x1 = x2✱ y1 = y2 t❤❡ ♣♦✐♥tsA1 ❛♥❞ A2 ❝♦✐♥❝✐❞❡ ❛♥❞ t❤❡ ❢♦r♠✉❧❛ (∗) ②✐❡❧❞s d = 0✳

❆s ❛♥ ❡①❡r❝✐s❡✱ ❧❡t ✉s ✜♥❞ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ❝❡♥tr❡ ♦❢ ❛ ❝✐r❝❧❡ ❝✐r❝✉♠✲s❝r✐❜❡❞ ❛❜♦✉t ❛ tr✐❛♥❣❧❡ ✇✐t❤ t❤❡ ✈❡rt✐❝❡s (x1, y1)✱ (x2, y2)✱ ❛♥❞ (x3, y3)✳

▲❡t (x, y)✱ ❜❡ t❤❡ ❝❡♥tr❡ ♦❢ t❤❡ ❝✐r❝✉♠❝✐r❝❧❡✳ ❙✐♥❝❡ ✐t ✐s ❡q✉✐❞✐st❛♥t ❢r♦♠t❤❡ ✈❡rt✐❝❡s ♦❢ t❤❡ tr✐❛♥❣❧❡✱ ✇❡ ❞❡r✐✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥s ❢♦r t❤❡ r❡q✉✐r❡❞

✶✺✳✸✳ ❚❍❊ ❉■❙❚❆◆❈❊ ❇❊❚❲❊❊◆ P❖■◆❚❙ ✶✻✼

❋✐❣✉r❡ ✶✺✳✽✿ ❉✐st❛♥❝❡ ♦❢ t✇♦ ♣♦✐♥ts

❋✐❣✉r❡ ✶✺✳✾✿ ❉✐st❛♥❝❡ ♦❢ t✇♦ s♣❡❝✐❛❧ ♣♦✐♥ts

✶✻✽ ❈❍❆P❚❊❘ ✶✺✳ ❈❆❘❚❊❙■❆◆ ❈❖❖❘❉■◆❆❚❊❙ ■◆ ❆ P▲❆◆❊

❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ❝❡♥tr❡ ♦❢ t❤❡ ❝✐r❝❧❡ ✭x ❛♥❞ y✮✳ ❚❤✉s✱ ✇❡ ❤❛✈❡

(x− x1)2 + (y − y1)

2 = (x− x2)2 + (y − y2)

2,

(x− x1)2 + (y − y1)

2 = (x− x3)2 + (y − y3)

2,

♦r ❛❢t❡r ♦❜✈✐♦✉s tr❛♥s❢♦r♠❛t✐♦♥s

2(x2 − x1)x+ 2(y2 − y1)y = x2

2 + y22 − x2

1 − y21,

2(x3 − x1)x+ 2(y3 − y1)y = x2

3 + y23 − x2

1 − y21.

❚❤✉s✱ ✇❡ ❤❛✈❡ ❛ s②st❡♠ ♦❢ t✇♦ ❧✐♥❡❛r ❡q✉❛t✐♦♥s ❢♦r ❞❡t❡r♠✐♥✐♥❣ t❤❡ ✉♥✲❦♥♦✇♥s x ❛♥❞ y✳

✶✺✳✹ ❊①❡r❝✐s❡s

✶✳ ❋✐♥❞ ♦♥ t❤❡ x✲❛①✐s t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ❛ ♣♦✐♥t ❡q✉✐❞✐st❛♥t ❢r♦♠ t❤❡ t✇♦❣✐✈❡♥ ♣♦✐♥ts A(x1, y1)✱ ❛♥❞ B(x2, y2)✳ ❈♦♥s✐❞❡r t❤❡ ❝❛s❡ A(0, a)✱ B(b, 0)✳

✷✳ ●✐✈❡♥ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t✇♦ ✈❡rt✐❝❡s A ❛♥❞ B ♦❢ ❛♥ ❡q✉✐❧❛t❡r❛❧ tr✐❛♥❣❧❡ABC✳ ❍♦✇ t♦ ✜♥❞ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ t❤✐r❞ ✈❡rt❡①❄ ❈♦♥s✐❞❡r t❤❡ ❝❛s❡A(0, a)✱ B(a, 0)✳

✸✳ ●✐✈❡♥ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t✇♦ ❛❞❥❛❝❡♥t ✈❡rt✐❝❡s A ❛♥❞ B ♦❢ ❛ sq✉❛r❡ABCD✳ ❍♦✇ ❛r❡ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ r❡♠❛✐♥✐♥❣ ✈❡rt✐❝❡s ❢♦✉♥❞❄ ❈♦♥s✐❞❡rt❤❡ ❝❛s❡ A(a, 0)✱ B(0, b)✳

✹✳ ❲❤❛t ❝♦♥❞✐t✐♦♥ ♠✉st ❜❡ s❛t✐s✜❡❞ ❜② t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ✈❡rt✐❝❡s ♦❢❛ tr✐❛♥❣❧❡ ABC s♦ ❛s t♦ ♦❜t❛✐♥ ❛ r✐❣❤t✲❛♥❣❧❡❞ tr✐❛♥❣❧❡ ✇✐t❤ ❛ r✐❣❤t ❛♥❣❧❡ ❛tt❤❡ ✈❡rt❡① C❄

✺✳ ❲❤❛t ❝♦♥❞✐t✐♦♥ ♠✉st ❜❡ s❛t✐s✜❡❞ ❜② t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ✈❡rt✐❝❡s ♦❢❛ tr✐❛♥❣❧❡ ABC s♦ t❤❛t t❤❡ ❛♥❣❧❡ A ❡①❝❡❡❞s t❤❡ ❛♥❣❧❡ B❄

✻✳ ❆ q✉❛❞r✐❧❛t❡r❛❧ ABCD ✐s s♣❡❝✐✜❡❞ ❜② t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ✐ts ✈❡rt✐❝❡s✳❍♦✇ t♦ ✜♥❞ ♦✉t ✇❤❡t❤❡r ♦r ♥♦t ✐s ✐t ✐♥s❝r✐❜❡❞ ✐♥ ❛ ❝✐r❝❧❡❄

✼✳ Pr♦✈❡ t❤❛t ❢♦r ❛♥② r❡❛❧ a✱ a1✱ a2✱ b✱ b1✱ b2 t❤❡r❡ ❤♦❧❞s t❤❡ ❢♦❧❧♦✇✐♥❣✐♥❡q✉❛❧✐t②

√

(a1 − a)2 + (b1 − b)2 +√

(a2 − a)2 + (b2 − b)2 ≥√

(a1 − a2)2 + (b1 − b2)2.

❚♦ ✇❤❛t ❣❡♦♠❡tr✐❝❛❧ ❢❛❝t ❞♦❡s ✐t ❝♦rr❡s♣♦♥❞❄

✶✺✳✺✳ ❉■❱■❉■◆● ❆ ▲■◆❊ ❙❊●▼❊◆❚ ■◆ ❆ ●■❱❊◆ ❘❆❚■❖ ✶✻✾

❋✐❣✉r❡ ✶✺✳✶✵✿ ❉✐✈✐❞✐♥❣ ❛ ❧✐♥❡ s❡❣♠❡♥t

✶✺✳✺ ❉✐✈✐❞✐♥❣ ❛ ❧✐♥❡ s❡❣♠❡♥t ✐♥ ❛ ❣✐✈❡♥ r❛t✐♦

▲❡t t❤❡r❡ ❜❡ ❣✐✈❡♥ t✇♦ ❞✐✛❡r❡♥t ♣♦✐♥ts ♦♥ t❤❡ xy✲♣❧❛♥❡✿ A1(x1, y1) ❛♥❞A2(x2, y2)✳ ❋✐♥❞ t❤❡ ❝♦♦r❞✐♥❛t❡s x ❛♥❞ y ♦❢ t❤❡ ♣♦✐♥t A ✇❤✐❝❤ ❞✐✈✐❞❡s t❤❡s❡❣♠❡♥t A1A2 ✐♥ t❤❡ r❛t✐♦ λ1 : λ2✳

❙✉♣♣♦s❡ t❤❡ s❡❣♠❡♥t A1A2 ✐s ♥♦t ♣❛r❛❧❧❡❧ t♦ t❤❡ x✲❛①✐s✳ Pr♦❥❡❝t✐♥❣ t❤❡♣♦✐♥ts A1✱ A✱ A2 ♦♥ t❤❡ y✲❛①✐s✱ ✇❡ ❤❛✈❡ ✭❋✐❣✳ ✶✺✳✶✵✮

A1A

AA2

=A1A

AA2

=λ1

λ2

.

❙✐♥❝❡ t❤❡ ♣♦✐♥ts A1✱ A2✱ A ❤❛✈❡ t❤❡ s❛♠❡ ♦r❞✐♥❛t❡s ❛s t❤❡ ♣♦✐♥ts A1✱ A2✱A✱ r❡s♣❡❝t✐✈❡❧②✱ ✇❡ ❣❡t

A1A = |y1 − y|, AA2 = |y − y2|.

❈♦♥s❡q✉❡♥t❧②✱|y1 − y||y − y2|

=λ1

λ2

.

❙✐♥❝❡ t❤❡ ♣♦✐♥t A ❧✐♥❡s ❜❡t✇❡❡♥ A1 ❛♥❞ A2✱ y1 − y ❛♥❞ y− y2 ❤❛✈❡ t❤❡ s❛♠❡s✐❣♥✳

❚❤❡r❡❢♦r❡|y1 − y||y − y2|

=y1 − y

y − y2=

λ1

λ2

.

❲❤❡♥❝❡ ✇❡ ✜♥❞

y =λ2y1 + λ1y2λ1 + λ2

. (∗)

✶✼✵ ❈❍❆P❚❊❘ ✶✺✳ ❈❆❘❚❊❙■❆◆ ❈❖❖❘❉■◆❆❚❊❙ ■◆ ❆ P▲❆◆❊

■❢ t❤❡ s❡❣♠❡♥t A1A2 ✐s ♣❛r❛❧❧❡❧ t♦ t❤❡ x✲❛①✐s✱ t❤❡♥

y1 = y2 = y.

❚❤❡ s❛♠❡ r❡s✉❧t ✐s ②✐❡❧❞❡❞ ❜② t❤❡ ❢♦r♠✉❧❛ (∗) ✇❤✐❝❤ ✐s t❤✉s tr✉❡ ❛♥②♣♦s✐t✐♦♥s ♦❢ t❤❡ ♣♦✐♥ts A1 ❛♥❞ A2✳

❚❤❡ ❛❜s❝✐ss❛ ♦❢ t❤❡ ♣♦✐♥t A ✐s ❢♦✉♥❞ ❛♥❛❧♦❣♦✉s❧②✳ ❋♦r ✐t ✇❡ ❣❡t t❤❡ ❢♦r♠✉❧❛

x =λ2x1 + λ1x2

λ1 + λ2

.

❲❡ ♣✉t λ1

λ1+λ2= t✳ ❚❤❡♥ λ2

λ1+λ2= 1− t✳

❈♦♥s❡q✉❡♥t❧②✱ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ❛♥② ♣♦✐♥t C ♦❢ ❛ s❡❣♠❡♥t ✇✐t❤ t❤❡ ❡♥❞✲♣♦✐♥ts A(x1, y1) ❛♥❞ B(x2, y2) ♠❛② ❜❡ r❡♣r❡s❡♥t❡❞ ❛t ❢♦❧❧♦✇s

x = (1− t)x1 + tx2, y = (1− t)y1 + ty2, 0 ≤ t ≤ 1.

▲❡t ✉s ✜♥❞ ❧♦❝❛t✐♦♥ ♦❢ ♣♦✐♥ts C(x, y) ❢♦r t < 0 ❛♥❞ t > 1✳ ❚♦ ❞♦ t❤✐s ✐♥❝❛s❡ ♦❢ t < 0 ✇❡ s♦❧✈❡ ♦✉r ❢♦r♠✉❧❛s ✇✐t❤ r❡s♣❡❝t t♦ x1✱ y1✳ ❲❡ ❣❡t

x1 =1 · x+ (−t)x2

1− t,

y1 =1 · y + (−t)y2

1− t.

❍❡♥❝❡✱ ✐t ✐s ❝❧❡❛r t❤❛t t❤❡ ♣♦✐♥t A(x1, y1) ✐s s✐t✉❛t❡❞ ♦♥ t❤❡ ❧✐♥❡ s❡❣♠❡♥t CB❛♥❞ ❞✐✈✐❞❡s t❤✐s s❡❣♠❡♥t ✐♥ t❤❡ r❛t✐♦ (−t) : 1✳ ❚❤✉s✱ ❢♦r t < 0 ♦✉r ❢♦r♠✉❧❛s②✐❡❧❞ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ♣♦✐♥t ❧②✐♥❣ ♦♥ t❤❡ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ s❡❣♠❡♥t AB❜❡②♦♥❞ t❤❡ ♣♦✐♥t A✳ ■t ✐s ♣r♦✈❡❞ ✐♥ ❛ s✐♠✐❧❛r ✇❛② t❤❛t ❢♦r t > 1 t❤❡ ❢♦r♠✉❧❛s②✐❡❧❞ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ♣♦✐♥t ❧♦❝❛t❡❞ ♦♥ t❤❡ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ s❡❣♠❡♥tAB ❜❡②♦♥❞ t❤❡ ♣♦✐♥t B✳

❆s ❛♥ ❡①❡r❝✐s❡✱ ❧❡t ✉s ♣r♦✈❡ ❈❡✈❛✬s t❤❡♦r❡♠ ❢r♦♠ ❡❧❡♠❡♥t❛r② ❣❡♦♠❡tr②✳■t st❛t❡s✿ ■t t❤❡ s✐❞❡s ♦❢ ❛ tr✐❛♥❣❧❡ ❛r❡ ❞✐✈✐❞❡❞ ✐♥ t❤❡ r❛t✐♦ a : b✱ c : a✱b : c✱ t❛❦❡♥ ✐♥ ♦r❞❡r ♦❢ ♠♦✈✐♥❣ r♦✉♥❞ t❤❡ tr✐❛♥❣❧❡ ✭s❡❡ ❋✐❣✳ ✶✺✳✶✶✮✱ t❤❡♥ t❤❡s❡❣♠❡♥ts ❥♦✐♥✐♥❣ t❤❡ ✈❡rt✐❝❡s ♦❢ t❤❡ tr✐❛♥❣❧❡ t♦ t❤❡ ♣♦✐♥ts ♦❢ ❞✐✈✐s✐♦♥ ♦❢ t❤❡♦♣♣♦s✐t❡ s✐❞❡s ✐♥t❡rs❡❝t ✐♥ ♦♥ ♦♥❡ ♣♦✐♥t✳

▲❡t A(x1, y1)✱ B(x2, y2)✱ ❛♥❞ C(x3, y3) ❜❡ t❤❡ ✈❡rt✐❝❡s ♦❢ t❤❡ tr✐❛♥❣❧❡ ❛♥❞A✱ B✱ C t❤❡ ♣♦✐♥ts ♦❢ ❞✐✈✐s✐♦♥ ♦❢ t❤❡ ♦♣♣♦s✐t❡ s✐❞❡s ✭❋✐❣✳ ✶✺✳✶✶✮✳ ❚❤❡ ❝♦♦r✲❞✐♥❛t❡s ♦❢ t❤❡ ♣♦✐♥t A ❛r❡✿

x =bx2 + cx3

b+ c, y =

by2 + cy3b+ c

,

✶✺✳✻✳ ❊❳❊❘❈■❙❊❙ ✶✼✶

❋✐❣✉r❡ ✶✺✳✶✶✿ ❈❡✈❛✬s t❤❡♦r❡♠

▲❡t ✉s ❞✐✈✐❞❡ t❤❡ ❧✐♥❡ s❡❣♠❡♥t AA ✐♥ t❤❡ r❛t✐♦ (b+c) : a✳ ❚❤❡♥ t❤❡ ❝♦♦r❞✐♥❛t❡s♦❢ t❤❡ ♣♦✐♥t ♦❢ ❞✐✈✐s✐♦♥ ✇✐❧❧ ❜❡

x =ax1 + bx2 + cx3

a+ b+ c,

y =ay1 + by2 + cy3

a+ b+ c.

■❢ t❤❡ s❡❣♠❡♥t BB ✐s ❞✐✈✐❞❡❞ ✐♥ t❤❡ r❛t✐♦ (a+ c) : b✱ t❤❡♥ ✇❡ ❣❡t t❤❡ s❛♠❡❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ♣♦✐♥t ♦❢ ❞✐✈✐s✐♦♥✳ ❚❤❡ s❛♠❡ ❝♦♦r❞✐♥❛t❡s ❛r❡ ♦❜t❛✐♥❡❞ ✇❤❡♥❞✐✈✐❞✐♥❣ t❤❡ s❡❣♠❡♥t CC ✐♥ t❤❡ r❛t✐♦ (a + b) : c✳ ❍❡♥❝❡✱ t❤❡ s❡❣♠❡♥ts AA✱BB✱ ❛♥❞ CC ❤❛✈❡ ❛ ♣♦✐♥t ✐♥ ❝♦♠♠♦♥✱ ✇❤✐❝❤ ✇❛s r❡q✉✐r❡❞ t♦ ❜❡ ♣r♦✈❡❞✳

▲❡t ✉s ♥♦t❡ ❤❡r❡ t❤❛t t❤❡ t❤❡♦r❡♠s ♦❢ ❡❧❡♠❡♥t❛r② ❣❡♦♠❡tr② ♦♥ ✐♥t❡rs❡❝t✐♥❣♠❡❞✐❛♥s✱ ❜✐s❡❝t♦rs✱ ❛♥❞ ❛❧t✐t✉❞❡s ✐♥ t❤❡ tr✐❛♥❣❧❡ ❛r❡ ♣❛rt✐❝✉❧❛r ❝❛s❡s ♦❢ ❈❡✈❛✬st❤❡♦r❡♠✳

✶✺✳✻ ❊①❡r❝✐s❡s

✶✳ ●✐✈❡♥ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤r❡❡ ✈❡rt✐❝❡s ♦❢ ❛ ♣❛r❛❧❧❡❧♦❣r❛♠✿ (x1, y1)✱(x2, y2)✱ ❛♥❞ (x3, y3)✳ ❋✐♥❞ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ❢♦✉rt❤ ✈❡rt❡① ❛♥❞ t❤❡❝❡♥tr♦✐❞✳

✷✳ ●✐✈❡♥ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ✈❡rt✐❝❡s ♦❢ ❛ tr✐❛♥❣❧❡✿ (x1, y1)✱ (x2, y2)✱❛♥❞ (x3, y3)✳ ❋✐♥❞ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ♣♦✐♥t ♦❢ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡ ♠❡❞✐❛♥s✳

✸✳ ●✐✈❡♥ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ♠✐❞✲♣♦✐♥ts ♦❢ t❤❡ s✐❞❡s ♦❢ ❛ tr✐❛♥❣❧❡(x1, y1)✱ (x2, y2)✱ ❛♥❞ (x3, y3)✳ ❋✐♥❞ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤✐s ✈❡rt✐❝❡s✳

✶✼✷ ❈❍❆P❚❊❘ ✶✺✳ ❈❆❘❚❊❙■❆◆ ❈❖❖❘❉■◆❆❚❊❙ ■◆ ❆ P▲❆◆❊

✹✳ ●✐✈❡♥ ❛ tr✐❛♥❣❧❡ ✇✐t❤ t❤❡ ✈❡rt✐❝❡s (x1, y1)✱ (x2, y2)✱ ❛♥❞ (x3, y3)✳ ❋✐♥❞t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ✈❡rt✐❝❡s ❛ ❤♦♠♦t❤❡t✐❝ tr✐❛♥❣❧❡ ✇✐t❤ t❤❡ r❛t✐♦ ♦❢ s✐♠✐✲❧❛r✐t② λ ❛♥❞ t❤❡ ❝❡♥tr❡ ♦❢ s✐♠✐❧✐t✉❞❡ ❛t ♣♦✐♥t (x0, y0)✳

✺✳ P♦✐♥t A ✐s s❛✐❞ t♦ ❞✐✈✐❞❡ t❤❡ ❧✐♥❡ s❡❣♠❡♥t A1A2 ❡①t❡r♥❛❧❧② ✐♥ t❤❡ r❛t✐♦λ1 : λ2 ✐❢ t❤✐s ♣♦✐♥t ❧✐❡s ♦♥ ❛ str❛✐❣❤t ❧✐♥❡ ❥♦✐♥✐♥❣ t❤❡ ♣♦✐♥ts A1 ❛♥❞ A2 ♦✉ts✐❞❡t❤❡ s❡❣♠❡♥t A1A2 ❛♥❞ t❤❡ r❛t✐♦ ♦❢ ✐ts ❞✐st❛♥❝❡s ❢r♦♠ t❤❡ ♣♦✐♥ts A1 ❛♥❞ A2

✐s ❡q✉❛❧ t♦ λ1 : λ2✳ ❙❤♦✇ t❤❛t t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ♣♦✐♥t A ❛r❡ ❡①♣r❡ss❡❞✐♥ t❡r♠s ♦❢ t❤❡ ❝♦♦r❞✐♥❛t❡s (x1, y1)✱ (x2, y2) ♦❢ t❤❡ ♣♦✐♥ts A1 ❛♥❞ A2 ❜② t❤❡❢♦r♠✉❧❛s

x =λ2x1 − λ1x2

λ2 − λ1

, y =λ2y1 − λ1y2λ2 − λ1

.

✻✳ ❚✇♦ ❧✐♥❡ s❡❣♠❡♥ts ❛r❡ s♣❡❝✐✜❡❞ ❜② t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡✐r ❡♥❞✲♣♦✐♥ts✳❍♦✇ ❝❛♥ ✇❡ ✜♥❞ ♦✉t✱ ✇✐t❤♦✉t ✉s✐♥❣ ❛ ❞r❛✇✐♥❣✱ ✇❤❡t❤❡r t❤❡ s❡❣♠❡♥ts ✐♥t❡rs❡❝t♦r ♥♦t❄

✼✳ ❚❤❡ ❝❡♥tr❡ ♦❢ ❣r❛✈✐t② ♦❢ t✇♦ ♠❛ss❡s µ1 ❛♥❞ µ2 s✐t✉❛t❡❞ ❛t ♣♦✐♥tsA1(x1, y1) ❛♥❞ A2(x2, y2) ✐s ❞❡✜♥❡❞ ❛s ❛ ♣♦✐♥t A ✇❤✐❝❤ ❞✐✈✐❞❡s t❤❡ s❡❣♠❡♥tA1A2 ✐♥ t❤❡ r❛t✐♦ µ2 : µ1✳

❚❤✉s✱ ✐ts ❝♦♦r❞✐♥❛t❡s ❛r❡✿

x =µ1x1 + µ2x2

µ1 + µ2

, y =µ1y1 + µ2y2µ1 + µ2

.

❚❤❡ ❝❡♥tr❡ ♦❢ ❣r❛✈✐t② ♦❢ n ♠❛ss❡s µi s✐t✉❛t❡❞ ❛t ♣♦✐♥ts Ai ✐s ❞❡t❡r♠✐♥❡❞ ❜②✐♥❞✉❝t✐♦♥✳ ■♥❞❡❡❞✱ ✐❢ A′

n ✐s t❤❡ ❝❡♥tr❡ ♦❢ ❣r❛✈✐t② ♦❢ t❤❡ ✜rst n−1 ♠❛ss❡s✱ t❤❡♥t❤❡ ❝❡♥tr❡ ♦❢ ❣r❛✈✐t② ♦❢ ❛❧❧ n ♠❛ss❡s ✐s ❞❡t❡r♠✐♥❡❞ ❛s t❤❡ ❝❡♥tr❡ ♦❢ ❣r❛✈✐t② ♦❢t✇♦ ♠❛ss❡s✿ µn ❧♦❝❛t❡❞ ❛t ♣♦✐♥t An✱ ❛♥❞ µ1 + · · · + µn−1✱ s✐t✉❛t❡❞ ❛t ♣♦✐♥tA′

n✳ ❲❡ t❤❡♥ ❞❡r✐✈❡ t❤❡ ❢♦r♠✉❧❛s ❢♦r t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ❝❡♥tr❡ ♦❢ ❣r❛✈✐t②♦❢ t❤❡ ♠❛ss❡s µi s✐t✉❛t❡❞ ❛t ♣♦✐♥ts Ai(xi, yi)✿

x =µ1x1 + · · ·+ µnxn

µ1 + · · ·+ µn

, y =µ1y1 + · · ·+ µnynµ1 + · · ·+ µn

✶✺✳✼ ❚❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ❝✐r❝❧❡

▲❡t t❤❡r❡ ❜❡ ❣✐✈❡♥ ❛ ❝✉r✈❡ ♦♥ t❤❡ xy✲♣❧❛♥❡ ✭❋✐❣✳ ✶✺✳✶✷✮✳ ❚❤❡ ❡q✉❛t✐♦♥ϕ(x, y) = 0 ✐s ❝❛❧❧❡❞ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ❝✉r✈❡ ✐♥ t❤❡ ✐♠♣❧✐❝✐t ❢♦r♠ ✐❢ ✐t ✐ss❛t✐s✜❡❞ ❜② t❤❡ ❝♦♦r❞✐♥❛t❡s (x, y) ♦❢ ❛♥② ♣♦✐♥t ♦❢ t❤✐s ❝✉r✈❡✳ ❆♥② ♣❛✐r ♦❢♥✉♠❜❡rs x✱ y✱ s❛t✐s❢②✐♥❣ t❤❡ ❡q✉❛t✐♦♥ ϕ(x, y) = 0 r❡♣r❡s❡♥ts t❤❡ ❝♦♦r❞✐♥❛t❡s♦❢ ❛ ♣♦✐♥t ♦♥ t❤❡ ❝✉r✈❡✳ ❆s ✐s ♦❜✈✐♦✉s✱ ❛ ❝✉r✈❡ ✐s ❞❡✜♥❡❞ ❜② ✐ts ❡q✉❛t✐♦♥✱t❤❡r❡❢♦r❡ ✇❡ ♠❛② s♣❡❛❦ ♦❢ r❡♣r❡s❡♥t✐♥❣ ❛ ❝✉r✈❡ ❜② ✐ts ❡q✉❛t✐♦♥✳

■♥ ❛♥❛❧②t✐❝ ❣❡♦♠❡tr② t✇♦ ♣r♦❜❧❡♠s ❛r❡ ♦❢t❡♥ ❝♦♥s✐❞❡r❡❞✿ ✭✶✮ ❣✐✈❡♥ t❤❡❣❡♦♠❡tr✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ ❛ ❝✉r✈❡✱ ❢♦r♠ ✐ts ❡q✉❛t✐♦♥✿ ✭✷✮ ❣✐✈❡♥ t❤❡ ❡q✉❛t✐♦♥

✶✺✳✼✳ ❚❍❊ ❊◗❯❆❚■❖◆ ❖❋ ❆ ❈■❘❈▲❊ ✶✼✸

❋✐❣✉r❡ ✶✺✳✶✷✿ ❊q✉❛t✐♦♥ ♦❢ ❛ ❝✉r✈❡

❋✐❣✉r❡ ✶✺✳✶✸✿ ❊q✉❛t✐♦♥ ♦❢ ❛ ❝✐r❝❧❡

♦❢ ❛ ❝✉r✈❡✱ ✜♥❞ ♦✉t ✐ts ❣❡♦♠❡tr✐❝❛❧ ♣r♦♣❡rt✐❡s✳ ▲❡t ✉s ❝♦♥s✐❞❡r t❤❡s❡ ♣r♦❜❧❡♠s❛s ❛♣♣❧✐❡❞ t♦ t❤❡ ❝✐r❝❧❡ ✇❤✐❝❤ ✐s t❤❡ s✐♠♣❧❡st ❝✉r✈❡✳

❙✉♣♣♦s❡ t❤❛t A0(x0, y0) ✐s ❛♥ ❛r❜✐tr❛r② ♣♦✐♥t ♦❢ t❤❡ xy✲♣❧❛♥❡✱ ❛♥❞ R ✐s❛♥② ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ▲❡t ✉s ❢♦r♠ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ❝✐r❝❧❡ ✇✐t❤ ❝❡♥tr❡ A0 ❛♥❞r❛❞✐✉s R ✭❋✐❣✳ ✶✺✳✶✸✮✳

▲❡t A(x, y) ❜❡ ❛♥ ❛r❜✐tr❛r② ♣♦✐♥t ♦❢ t❤❡ ❝✐r❝❧❡✳ ■ts ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ ❝❡♥tr❡A0 ✐s ❡q✉❛❧ t♦ R✳ ❆❝❝♦r❞✐♥❣ t♦ ❙❡❝t✐♦♥ ✶✺✳✸✱ t❤❡ sq✉❛r❡ ♦❢ t❤❡ ❞✐st❛♥❝❡ ♦❢ t❤❡♣♦✐♥t A ❢r♦♠ A0 ✐s ❡q✉❛❧ t♦ (x − x0)

2 + (y − y0)2✳ ❚❤✉s✱ t❤❡ ❝♦♦r❞✐♥❛t❡s x✱

y ♦❢ ❛♥② ♣♦✐♥t A ♦❢ t❤❡ ❝✐r❝❧❡ s❛t✐s❢② t❤❡ ❡q✉❛t✐♦♥

(x− x0)2 + (y − y0)

2 −R2 = 0. (∗)

❈♦♥✈❡rs❡❧②✱ ❛♥② ♣♦✐♥t A ✇❤♦s❡ ❝♦♦r❞✐♥❛t❡s s❛t✐s❢② t❤❡ ❡q✉❛t✐♦♥ (∗) ❜❡✲

✶✼✹ ❈❍❆P❚❊❘ ✶✺✳ ❈❆❘❚❊❙■❆◆ ❈❖❖❘❉■◆❆❚❊❙ ■◆ ❆ P▲❆◆❊

❧♦♥❣s t♦ t❤❡ ❝✐r❝❧❡✱ s✐♥❝❡ ✐ts ❞✐st❛♥❝❡ ❢r♦♠ A0 ✐s ❡q✉❛❧ t♦ R✳■♥ ❝♦♥❢♦r♠✐t② ✇✐t❤ t❤❡ ❛❜♦✈❡ ❞❡✜♥✐t✐♦♥✱ t❤❡ ❡q✉❛t✐♦♥ (∗) ✐s ❛♥ ❡q✉❛t✐♦♥

♦❢ ❛ ❝✐r❝❧❡ ✇✐t❤ ❝❡♥tr❡ A0 ❛♥❞ r❛❞✐✉s R✳❲❡ ♥♦✇ ❝♦♥s✐❞❡r t❤❡ s❡❝♦♥❞ ♣r♦❜❧❡♠ ❢♦r t❤❡ ❝✉r✈❡ ❣✐✈❡♥ ❜② t❤❡ ❡q✉❛t✐♦♥

x2 + y2 + 2ax+ 2by + c = 0 (a2 + b2 − c > 0).

❚❤✐s ❡q✉❛t✐♦♥ ❝❛♥ ❜❡ r❡✇r✐tt❡♥ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉✐✈❛❧❡♥t ❢♦r♠✿

(x+ a)2 + (y + b)2 − (√a2 + b2 − c )2 = 0.

❲❤❡♥❝❡ ✐t ✐s s❡❡♥ t❤❛♥ ❛♥② ♣♦✐♥t (x, y) ♦❢ t❤❡ ❝✉r✈❡ ✐s ❢♦✉♥❞ ❛t ♦♥❡ ❛♥❞ t❤❡s❛♠❡ ❞✐st❛♥❝❡ ❡q✉❛❧ t♦

√a2 + b2 − c ❢r♦♠ t❤❡ ♣♦✐♥t (−a,−b)✱ ❛♥❞✱ ❤❡♥❝❡✱ t❤❡

❝✉r✈❡ ✐s ❛ ❝✐r❝❧❡ ✇✐t❤ ❝❡♥tr❡ (−a,−b) ❛♥❞ r❛❞✐✉s√a2 + b2 − c✳

▲❡t ✉s ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦❜❧❡♠ ❛s ❛♥ ❡①❛♠♣❧❡ ✐❧❧✉str❛t✐♥❣ t❤❡ ❛♣✲♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ ♠❡t❤♦❞ ♦❢ ❛♥❛❧②t✐❝ ❣❡♦♠❡tr②✿ ❋✐♥❞ t❤❡ ❧♦❝✉s ♦❢ ♣♦✐♥ts ✐♥ ❛♣❧❛♥❡ t❤❡ r❛t✐♦ ♦❢ ✇❤♦s❡ ❞✐st❛♥❝❡s ❢r♦♠ t✇♦ ❣✐✈❡♥ ♣♦✐♥ts A ❛♥❞ B ✐s ❝♦♥st❛♥t❛♥❞ ✐s ❡q✉❛❧ t♦ k 6= 1✳ ✭❚❤❡ ❧♦❝✉s ✐s ❞❡✜♥❡❞ ❛s ❛ ✜❣✉r❡ ✇❤✐❝❤ ❝♦♥s✐sts ♦❢❛❧❧ t❤❡ ♣♦✐♥ts ♣♦ss❡ss✐♥❣ t❤❡ ❣✐✈❡♥ ❣❡♦♠❡tr✐❝❛❧ ♣r♦♣❡rt②✳ ■♥ t❤❡ ❝❛s❡ ✉♥❞❡r❝♦♥s✐❞❡r❛t✐♦♥ ✇❡ s♣❡❛❦ ♦❢ ❛ s❡t ♦❢ ❛❧❧ t❤❡ ♣♦✐♥ts ✐♥ t❤❡ ♣❧❛♥❡ ❢♦r ✇❤✐❝❤ t❤❡r❛t✐♦ ♦❢ t❤❡ ❞✐st❛♥❝❡s ❢r♦♠ t❤❡ t✇♦ ♣♦✐♥ts A ❛♥❞ B ✐s ❝♦♥st❛♥t✮✳

❙✉♣♣♦s❡ t❤❛t 2a ✐s t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ♣♦✐♥ts A ❛♥❞ B✳ ❲❡ t❤❡♥✐♥tr♦❞✉❝❡ ❛ r❡❝t❛♥❣✉❧❛r ❈❛rt❡s✐❛♥ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ♦♥ t❤❡ ♣❧❛♥❡ t❛❦✐♥❣ t❤❡str❛✐❣❤t ❧✐♥❡ AB ❢♦r t❤❡ x✲❛①✐s ❛♥❞ t❤❡ ♠✐❞♣♦✐♥t ♦❢ t❤❡ s❡❣♠❡♥t AB ❢♦r t❤❡♦r✐❣✐♥✳ ▲❡t✱ ❢♦r ❞❡✜♥✐t❡♥❡ss✱ t❤❡ ♣♦✐♥t A ❜❡ s✐t✉❛t❡❞ ♦♥ t❤❡ ♣♦s✐t✐✈❡ s❡♠✐✲❛①✐s x✳ ❚❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ♣♦✐♥t A ✇✐❧❧ t❤❡♥ ❜❡✿ x = a✱ y = 0✱ ❛♥❞ t❤❡❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ♣♦✐♥t B ✇✐❧❧ ❜❡✿ x = −a✱ y = 0✳ ▲❡t (x, y) ❜❡ ❛♥ ❛r❜✐tr❛r②♣♦✐♥t ♦❢ t❤❡ ❧♦❝✉s✳ ❚❤❡ sq✉❛r❡s ♦❢ ✐ts ❞✐st❛♥❝❡s ❢r♦♠ t❤❡ ♣♦✐♥ts A ❛♥❞ B ❛r❡r❡s♣❡❝t✐✈❡❧② ❡q✉❛❧ t♦ (x − a)2 + y2 ❛♥❞ (x + a)2 + y2✳ ❚❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡❧♦❝✉s ✐s

(x− a)2 + y2

(x+ a)2 + y2= k2,

♦r

x2 + y2 +2(k2 + 1)

k2 − 1ax+ a2 = 0.

❚❤❡ ❧♦❝✉s r❡♣r❡s❡♥ts ❛ ❝✐r❝❧❡ ✭❝❛❧❧❡❞ ❆♣♦❧❧♦♥✐✉s✬ ❝✐r❝❧❡✮✳

✶✺✳✽ ❊①❡r❝✐s❡s

✶✺✳✾✳ ❚❍❊ ❊◗❯❆❚■❖◆ ❖❋ ❆ ❈❯❘❱❊ ❘❊P❘❊❙❊◆❚❊❉ ❇❨ P❆❘❆▼❊❚❊❘❙✶✼✺

✶✳ ❲❤❛t ♣❡❝✉❧✐❛r✐t✐❡s ✐♥ t❤❡ ♣♦s✐t✐♦♥ ♦❢ t❤❡ ❝✐r❝❧❡

x2 + y2 + 2ax+ 2by + c = 0 (a2 + b2 − c > 0)

r❡❧❛t✐✈❡ t♦ t❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ t❛❦❡ ♣❧❛❝❡ ✐❢

(1) a = 0; (2) b = 0; (3) c = 0;

(4) a = 0, b = 0; (5) a = 0, c = 0; (6) b = 0, c = 0?

✷✳ ❙❤♦✇ t❤❛t ✐❢ ✇❡ s✉❜st✐t✉t❡ ✐♥ t❤❡ ❧❡❢t✲❤❛♥❞ ♠❡♠❜❡r ♦❢ t❤❡ ❡q✉❛t✐♦♥ ♦❢❛ ❝✐r❝❧❡ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ❛♥② ♣♦✐♥t ❧②✐♥❣ ♦✉ts✐❞❡ t❤❡ ❝✐r❝❧❡✱ t❤❡♥ t❤❡ sq✉❛r❡♦❢ t❤❡ ❧❡♥❣t❤ ♦❢ ❛ t❛♥❣❡♥t ❞r❛✇♥ ❢r♦♠ t❤✐s ♣♦✐♥t t♦ t❤❡ ❝✐r❝❧❡ ✐s ♦❜t❛✐♥❡❞✳

✸✳ ❚❤❡ ♣♦✇❡r ♦❢ ❛ ♣♦✐♥t A ✇✐t❤ r❡❢❡r❡♥❝❡ t♦ ❛ ❝✐r❝❧❡ ✐s ❞❡✜♥❡❞ ❛s t❤❡♣r♦❞✉❝t ♦❢ t❤❡ s❡❣♠❡♥ts ♦❢ ❛ s❡❝❛♥t ❞r❛✇♥ t❤r♦✉❣❤ t❤❡ ♣♦✐♥t A t❛❦❡♥ ✇✐t❤♣❧✉s ❢♦r ♦✉ts✐❞❡ ♣♦✐♥ts ❛♥❞ ✇✐t❤ ♠✐♥✉s ❢♦r ✐♥s✐❞❡ ♣♦✐♥ts✳ ❙❤♦✇ t❤❛t t❤❡ ❧❡❢t✲❤❛♥❞ ♠❡♠❜❡r ♦❢ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ❝✐r❝❧❡ x2 + y2 + 2ax + 2by + c = 0 ❣✐✈❡st❤❡ ♣♦✇❡r ♦❢ t❤✐s ♣♦✐♥t ✇✐t❤ r❡❢❡r❡♥❝❡ t♦ ❛ ❝✐r❝❧❡ ✇❤❡♥ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ❛♥❛r❜✐tr❛r② ♣♦✐♥t ❛r❡ s✉❜st✐t✉t❡❞ ✐♥ ✐t✳

✹✳ ❋♦r♠ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❧♦❝✉s ♦❢ ♣♦✐♥ts ♦❢ t❤❡ xy✲♣❧❛♥❡ t❤❡ s✉♠ ♦❢✇❤♦s❡ ❞✐st❛♥❝❡s ❢r♦♠ t✇♦ ❣✐✈❡♥ ♣♦✐♥ts F1(c, 0) ❛♥❞ F2(−c, 0) ✐s ❝♦♥st❛♥t ❛♥❞✐s ❡q✉❛❧ t♦ 2a ✭t❤❡ ❡❧❧✐♣s❡✮✳ ❙❤♦✇ t❤❛t t❤❡ ❡q✉❛t✐♦♥ ✐s r❡❞✉❝❡❞ t♦ t❤❡ ❢♦r♠x2

a2+ y2

b2= 1✱ ✇❤❡r❡ b2 = a2 − c2✳

✺✳ ❋♦r♠ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❧♦❝✉s ♦❢ ♣♦✐♥ts ♦❢ t❤❡ xy✲♣❧❛♥❡ t❤❡ ❞✐✛❡r❡♥❝❡♦❢ ✇❤♦s❡ ❞✐st❛♥❝❡s ❢r♦♠ t✇♦ ❣✐✈❡♥ ♣♦✐♥ts F1(c, 0) ❛♥❞ F2(−c, 0) ✐s ❝♦♥st❛♥t❛♥❞ ✐s ❡q✉❛❧ t♦ 2a ✭t❤❡ ❤②♣❡r❜♦❧❛✮✳ ❙❤♦✇ t❤❛t t❤❡ ❡q✉❛t✐♦♥ ✐s r❡❞✉❝❡❞ t♦ t❤❡❢♦r♠ x2

a2+ y2

b2= 1 ✇❤❡r❡ b2 = c2 − a2✳

✻✳ ❋♦r♠ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❧♦❝✉s ♦❢ ♣♦✐♥ts ♦❢ t❤❡ xy✲♣❧❛♥❡ ✇❤✐❝❤ ❛r❡❡q✉✐❞✐st❛♥t ❢r♦♠ t❤❡ ♣♦✐♥t F (0, p) ❛♥❞ t❤❡ x✲❛①✐s ✭t❤❡ ♣❛r❛❜♦❧❛✮✳

✶✺✳✾ ❚❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ❝✉r✈❡ r❡♣r❡s❡♥t❡❞ ❜②

♣❛r❛♠❡t❡rs

❙✉♣♣♦s❡ ❛ ♣♦✐♥t A ♠♦✈❡s ❛❧♦♥❣ ❛ ❝✉r✈❡✱ ❛♥❞ ❜② t❤❡ t✐♠❡ t ✐ts ❝♦♦r❞✐♥❛t❡s❛r❡✿ x = ϕ(t) ❛♥❞ y = ϕ(t)✳ ❆ s②st❡♠ ♦❢ ❡q✉❛t✐♦♥s

x = ϕ(t), y = ϕ(t),

s♣❡❝✐❢②✐♥❣ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ❛♥ ❛r❜✐tr❛r② ♣♦✐♥t ♦♥ t❤❡ ❝✉r✈❡ ❛s ❢✉♥❝t✐♦♥s t❤❡♣❛r❛♠❡t❡r t ✐s ❝❛❧❧❡❞ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ❝✉r✈❡ ✐♥ ♣❛r❛♠❡tr✐❝ ❢♦r♠✳

❚❤❡ ♣❛r❛♠❡t❡r t ✐s ♥♦t ♥❡❝❡ss❛r✐❧② t✐♠❡✱ ✐t ♠❛② ❜❡ ❛♥② ♦t❤❡r q✉❛♥t✐t②❝❤❛r❛❝t❡r✐③✐♥❣ t❤❡ ♣♦s✐t✐♦♥ ♦❢ ❛ ♣♦✐♥t ♦♥ t❤❡ ❝✉r✈❡✳

✶✼✻ ❈❍❆P❚❊❘ ✶✺✳ ❈❆❘❚❊❙■❆◆ ❈❖❖❘❉■◆❆❚❊❙ ■◆ ❆ P▲❆◆❊

❋✐❣✉r❡ ✶✺✳✶✹✿ ❉✐st❛♥❝❡ ♦❢ t✇♦ ♣♦✐♥ts

▲❡t ✉s ♥♦✇ ❢♦r♠ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ❝✐r❝❧❡ ✐♥ ♣❛r❛♠❡tr✐❝ ❢♦r♠✳❙✉♣♣♦s❡ t❤❡ ❝❡♥tr❡ ♦❢ ❛ ❝✐r❝❧❡ ✐s s✐t✉❛t❡❞ ❛t t❤❡ ♦r✐❣✐♥✱ ❛♥❞ t❤❡ r❛❞✐✉s ✐s

❡q✉❛❧ t♦ R✳ ❲❡ s❤❛❧❧ ❝❤❛r❛❝t❡r✐③❡ t❤❡ ♣♦s✐t✐♦♥ ♦❢ ♣♦✐♥t A ♦♥ t❤❡ ❝✐r❝❧❡ ❜② t❤❡❛♥❣❧❡ α ❢♦r♠❡❞ ❜② t❤❡ r❛❞✐✉s OA ✇✐t❤ t❤❡ ♣♦s✐t✐✈❡ s❡♠✐✲❛①✐s x ✭❋✐❣✳ ✶✺✳✶✹✮✳❆s ✐s ♦❜✈✐♦✉s✱ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ♣♦✐♥t A ❛r❡ ❡q✉❛❧ t♦ R cosα✱ R sinα✱❛♥❞✱ ❝♦♥s❡q✉❡♥t❧②✱ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❝✐r❝❧❡ ❤❛s s✉❝❤ ❛ ❢♦r♠✿

x = R cosα, y = R sinα.

❍❛✈✐♥❣ ❛♥ ❡q✉❛t✐♦♥ ♦❢ ❛ ❝✉r✈❡ ✐♥ ♣❛r❛♠❡tr✐❝ ❢♦r♠✿

x = ϕ(t), y = ϕ(t), (∗)

✇❡ ❝❛♥ ♦❜t❛✐♥ ✐ts ❡q✉❛t✐♦♥ ✐♥ ✐♠♣❧✐❝✐t ❢♦r♠✿

f(x, y) = 0.

❚♦ t❤✐s ❡✛❡❝t ✐t ✐s s✉✣❝✐❡♥t t♦ ❡❧✐♠✐♥❛t❡ t❤❡ ♣❛r❛♠❡t❡r t ❢r♦♠ t❤❡ ❡q✉❛t✐♦♥s(∗)✱ ✜♥❞✐♥❣ ♦♥❡ ❡q✉❛t✐♦♥ ❛♥❞ s✉❜st✐t✉t✐♥❣ ✐♥t♦ t❤❡ ♦t❤❡r✱ ♦r ✉s✐♥❣ ❛♥♦t❤❡r♠❡t❤♦❞✳

❋♦r ✐♥st❛♥❝❡✱ t♦ ❣❡t t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ❝✐r❝❧❡ r❡♣r❡s❡♥t❡❞ ❜② ❡q✉❛t✐♦♥s ✐♥♣❛r❛♠❡tr✐❝ ❢♦r♠ ✭✐✳❡✳ ✐♠♣❧✐❝✐t❧②✮ ✐t ✐s s✉✣❝✐❡♥t t♦ sq✉❛r❡ ❜♦t❤ ❡q✉❛❧✐t✐❡s ❛♥❞❛❞❞ t❤❡♥ t❡r♠✇✐s❡✳ ❲❡ t❤❡♥ ♦❜t❛✐♥ t❤❡ ❢❛♠✐❧✐❛r ❡q✉❛t✐♦♥ x2 + y2 = R2✳

❚❤❡ ❡❧✐♠✐♥❛t✐♦♥ ♦❢ t❤❡ ♣❛r❛♠❡t❡r ❢r♦♠ t❤❡ ❡q✉❛t✐♦♥s ♦❢ ❛ ❝✉r✈❡ r❡♣r❡✲s❡♥t❡❞ ♣❛r❛♠❡tr✐❝❛❧❧② ♥♦t ❛❧✇❛②s ②✐❡❧❞s ❛♥ ❡q✉❛t✐♦♥ ✐♥ ✐♠♣❧✐❝✐t ❢♦r♠ ✐♥ t❤❡s❡♥s❡ ♦❢ t❤❡ ❛❜♦✈❡ ❞❡✜♥✐t✐♦♥✳ ■t ♠❛♥② t✉r♥ ♦✉t t❤❛t ✐t ✐s s❛t✐s✜❡❞ ❜② t❤❡

✶✺✳✶✵✳ ❊❳❊❘❈■❙❊❙ ✶✼✼

♣♦✐♥ts ♥♦t ❜❡❧♦♥❣✐♥❣ t♦ t❤❡ ❝✉r✈❡✳ ■♥ t❤✐s ❝♦♥♥❡❝t✐♦♥ ❧❡t ✉s ❝♦♥s✐❞❡r t✇♦❡①❛♠♣❧❡s✳

❙✉♣♣♦s❡ ❛ ❝✉r✈❡ y ✐s ❣✐✈❡♥ ❜② t❤❡ ❡q✉❛t✐♦♥s ✐♥ ♣❛r❛♠❡tr✐❝ ❢♦r♠

x = a cos t, y = b sin t, 0 ≤ t ≤ 2π.

❉✐✈✐❞✐♥❣ t❤❡s❡ ❡q✉❛t✐♦♥s ❜② a ❛♥❞ b✱ r❡s♣❡❝t✐✈❡❧②✱ sq✉❛r✐♥❣ ❛♥❞ ❛❞❞✐♥❣ t❤❡♠t❡r♠✇✐s❡✱ ✇❡ ❣❡t t❤❡ ❡q✉❛t✐♦♥

x2

a2+

y2

b2= 1.

❚❤✐s ❡q✉❛t✐♦♥ ✐s ♦❜✈✐♦✉s❧② s❛t✐s✜❡❞ ❜② ❛❧❧ t❤❡ ♣♦✐♥ts ❜❡❧♦♥❣✐♥❣ t♦ t❤❡ ❝✉r✈❡y✳ ❈♦♥✈❡rs❡❧②✱ ✐❢ t❤❡ ♣♦✐♥t (x, y) s❛t✐s✜❡s t❤✐s ❡q✉❛t✐♦♥✱ t❤❡♥ t❤❡r❡ ❝❛♥ ❜❡❢♦✉♥❞ ❛♥ ❛♥❣❧❡ t ❢♦r ✇❤✐❝❤ x/a = cos t✱ y/b = sin t✱ ❛♥❞✱ ❝♦♥s❡q✉❡♥t❧②✱ ❛♥②♣♦✐♥t ♦❢ t❤❡ ♣❧❛♥❡ ✇❤✐❝❤ s❛t✐s✜❡s t❤✐s ❡q✉❛t✐♦♥✱ ❜❡❧♦♥❣s t♦ t❤❡ ❝✉r✈❡ y✳

▲❡t ♥♦✇ ❛ ❝✉r✈❡ y ❜❡ r❡♣r❡s❡♥t❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥s

x = cosh t, y = b sinh t, −∞ < t+∞,

✇❤❡r❡cosh t = (et + e−t)/2, sinh t = (et − e−t)/2.

❉✐✈✐❞✐♥❣ t❤❡s❡ ❡q✉❛t✐♦♥s ❜② a ❛♥❞ b✱ r❡s♣❡❝t✐✈❡❧②✱ ❛♥❞ t❤❡♥ sq✉❛r✐♥❣ t❤❡♠❛♥❞ s✉❜tr❛❝t✐♥❣ t❡r♠✇✐s❡✱ ✇❡ ❣❡t t❤❡ ❡q✉❛t✐♦♥

x2

a2− y2

b2= 1.

❚❤❡ ♣♦✐♥ts ♦❢ t❤❡ ❝✉r✈❡ y s❛t✐s❢② t❤✐s ❡q✉❛t✐♦♥✳ ❇✉t ♥♦t ❛♥② ♣♦✐♥t ✇❤✐❝❤s❛t✐s✜❡s t❤❡ ❡q✉❛t✐♦♥ ❜❡❧♦♥❣s t♦ y✳ ▲❡t ✉s✱ ❢♦r ✐♥st❛♥❝❡✱ ❝♦♥s✐❞❡r t❤❡ ♣♦✐♥t(−a, 0)✳ ❲❡ s❡❡ t❤❛t ✐t s❛t✐s✜❡s t❤❡ ❡q✉❛t✐♦♥✱ ❜✉t ❞♦❡s ♥♦t ❜❡❧♦♥❣ t♦ t❤❡❝✉r✈❡✱ s✐♥❝❡ ♦♥ t❤❡ ❝✉r✈❡ ya cosh t 6= −a✳

❙♦♠❡t✐♠❡s t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ❝✉r✈❡ r❡♣r❡s❡♥t❡❞ ✐♥ ✐♠♣❧✐❝✐t ❢♦r♠ ✐s ✉♥❞❡r✲st♦♦❞ ✐♥ ❛ ✇✐❞❡r ✇❛②✳ ❖♥❡ ❞♦❡s ♥♦t r❡q✉✐r❡ t❤❛t ❛♥② ♣♦✐♥t s❛t✐s❢②✐♥❣ t❤❡❡q✉❛t✐♦♥✱ ❜❡❧♦♥❣s t♦ t❤❡ ❝✉r✈❡✳

✶✺✳✶✵ ❊①❡r❝✐s❡s

✶✳ ❙❤♦✇ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥s ✐♥ ♣❛r❛♠❡tr✐❝ ❢♦r♠

x = R cos t+ a, y = R sin t+ b

✶✼✽ ❈❍❆P❚❊❘ ✶✺✳ ❈❆❘❚❊❙■❆◆ ❈❖❖❘❉■◆❆❚❊❙ ■◆ ❆ P▲❆◆❊

❋✐❣✉r❡ ✶✺✳✶✺✿ ❊①❡r❝✐s❡ ✸

r❡♣r❡s❡♥t ❛ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s R ✇✐t❤ ❝❡♥tr❡ ❛t ♣♦✐♥t (a, b)✳✷✳ ❋♦r♠ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ❝✉r✈❡ ❞❡s❝r✐❜❡❞ ❜② ❛ ♣♦✐♥t ♦♥ t❤❡ ❧✐♥❡ s❡❣♠❡♥t

♦❢ ❧❡♥❣t❤ a ✇❤❡♥ t❤❡ ❡♥❞✲♣♦✐♥ts ♦❢ t❤❡ s❡❣♠❡♥t s❧✐❞❡ ❛❧♦♥❣ t❤❡ ❝♦♦r❞✐♥❛t❡❛①❡s ✭t❤❡ s❡❣♠❡♥t ✐s ❞✐✈✐❞❡❞ ❜② t❤✐s ♣♦✐♥t ✐♥ t❤❡ r❛t✐♦ λ : µ✮✳ ❚❛❦❡ t❤❡ ❛♥❣❧❡❢♦r♠❡❞ ❜② t❤❡ s❡❣♠❡♥t ✇✐t❤ t❤❡ x✲❛①✐s ❢♦r t❤❡ ♣❛r❛♠❡t❡r✳ ❲❤❛t ✐s t❤❡ s❤❛♣❡♦❢ t❤❡ ❝✉r✈❡ ✐❢ λ : µ = 1❄

✸✳ ❆ tr✐❛♥❣❧❡ s❧✐❞❡s ❛❧♦♥❣ t❤❡ ❝♦♦r❞✐♥❛t❡ ❛①❡s ✇✐t❤ t✇♦ ♦❢ ✐ts ✈❡rt✐❝❡s✳❋♦r♠ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❝✉r✈❡ ❞❡s❝r✐❜❡❞ ❜② t❤❡ t❤✐r❞ ✈❡rt❡① ✭❋✐❣✳ ✶✺✳✶✺✮✳

✹✳ ❋♦r♠ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❝✉r✈❡ ❞❡s❝r✐❜❡❞ ❜② ❛ ♣♦✐♥t ♦♥ ❛ ❝✐r❝❧❡ ♦❢r❛❞✐✉s R ✇❤✐❝❤ r♦❧❧s ❛❧♦♥❣ t❤❡ x✲❛①✐s✳ ❋♦r t❤❡ ♣❛r❛♠❡t❡r t❛❦❡ t❤❡ ♣❛t❤ s❝♦✈❡r❡❞ ❜② t❤❡ ❝❡♥tr❡ ♦❢ t❤❡ ❝✐r❝❧❡ ❛♥❞ s✉♣♣♦s❡ t❤❛t ❛t t❤❡ ✐♥✐t✐❛❧ ♠♦♠❡♥t(s = 0) ♣♦✐♥t A ❝♦✐♥❝✐❞❡s ✇✐t❤ t❤❡ ♦r✐❣✐♥✳

✺✳ ❆ ❝✉r✈❡ ✐s ❣✐✈❡♥ ❜② t❤❡ ❡q✉❛t✐♦♥

ax2 + bxy + cy2 + dx+ ey = 0.

❙❤♦✇ t❤❛t✱ ❜② ✐♥tr♦❞✉❝✐♥❣ t❤❡ ♣❛r❛♠❡t❡r t = y/x✱ ✇❡ ❝❛♥ ♦❜t❛✐♥ t❤❡❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥s ♦❢ t❤✐s ❝✉r✈❡ ✐♥ ♣❛r❛♠❡tr✐❝ ❢♦r♠✿

x = − d+ et

a+ bt+ ct2,

y = − dt+ et2

a+ bt+ ct2.

❈❤❛♣t❡r ✶✻

❚❤❡ ❙tr❛✐❣❤t ▲✐♥❡

✶✻✳✶ ❚❤❡ ❣❡♥❡r❛❧ ❡q✉❛t✐♦♥ ♦❢ ❛ str❛✐❣❤t ❧✐♥❡

❚❤❡ str❛✐❣❤t ❧✐♥❡ ✐s t❤❡ s✐♠♣❧❡st ❛♥❞ ♠♦st ✇✐❞❡❧② ✉s❡❞ ❧✐♥❡✳❲❡ s❤❛❧❧ ♥♦✇ s❤♦✇ t❤❛t ❛♥② str❛✐❣❤t ❧✐♥❡ ❤❛s ❛♥ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❢♦r♠

ax+ by + c = 0, (∗)✇❤❡r❡ a✱ b✱ c ❛r❡ ❝♦♥st❛♥t✳ ❆♥❞ ❝♦♥✈❡rs❡❧②✱ ✐❢ a ❛♥❞ b ❛r❡ ♥♦t❤ ❜♦t❤ ③❡r♦✱t❤❡♥ t❤❡r❡ ❡①✐sts ❛ str❛✐❣❤t ❧✐♥❡ ❢♦r ✇❤✐❝❤ (∗) ✐s ✐ts ❡q✉❛t✐♦♥✳

▲❡t A1(a1, b1) ❛♥❞ A2(a2, b2) ❜❡ t✇♦ ❞✐✛❡r❡♥t ♣♦✐♥ts s✐t✉❛t❡❞ s②♠♠❡tr✐✲❝❛❧❧② ❛❜♦✉t ❛ ❣✐✈❡♥ str❛✐❣❤t ❧✐♥❡ ✭❋✐❣✳ ✶✻✳✶✮✳ ❚❤❡♥ ❛♥② ♣♦✐♥t A(x, y) ♦♥ t❤✐s❧✐♥❡ ✐s ❡q✉✐❞✐st❛♥t ❢r♦♠ t❤❡ ♣♦✐♥ts A1 ❛♥❞ A2✳ ❆♥❞ ❝♦♥✈❡rs❡❧②✱ ❛♥② ♣♦✐♥t A✇❤✐❝❤ ✐s ❡q✉✐❞✐st❛♥t ❢r♦♠ A1 ❛♥❞ A2 ❜❡❧♦♥❣s t♦ t❤❡ str❛✐❣❤t ❧✐♥❡✳ ❍❡♥❝❡✱ t❤❡❡q✉❛t✐♦♥ ♦❢ ❛ str❛✐❣❤t ❧✐♥❡ ✐s

(x− a1)2 + (y − b1)

2 = (x− a2)2 + (y − b2)

2.

❚r❛♥s♣♦s✐♥❣ ❛❧❧ t❡r♠s ♦❢ t❤❡ ❡q✉❛t✐♦♥ t♦ t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡✱ r❡♠♦✈✐♥❣ t❤❡sq✉❛r❡❞ ♣❛r❡♥t❤❡s❡s✱ ❛♥❞ ❝❛rr②✐♥❣ ♦✉t ♦❜✈✐♦✉s s✐♠♣❧✐✜❝❛t✐♦♥s✱ ✇❡ ❣❡t

2(a2 − a1)x+ 2(b2 − b1)y + (a21 + b21 − a22 − b22) = 0.

❚❤✉s✱ t❤❡ ✜rst ♣❛rt ♦❢ t❤❡ st❛t❡♠❡♥t ✐s ♣r♦✈❡❞✳❲❡ ♥♦✇ s❤❛❧❧ ♣r♦✈❡ t❤❡ s❡❝♦♥❞ ♣❛rt✳ ▲❡t B1 ❛♥❞ B2 ❜❡ t✇♦ ❞✐✛❡r❡♥t

♣♦✐♥ts ♦❢ t❤❡ xy✲♣❧❛♥❡ ✇❤♦s❡ ❝♦♦r❞✐♥❛t❡s s❛t✐s❢② t❤❡ ❡q✉❛t✐♦♥ (∗)✳ ❙✉♣♣♦s❡a1x+ b1y + c1 = 0

✐s t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ str❛✐❣❤t ❧✐♥❡ B1B2✳ ❚❤❡ s②st❡♠ ♦❢ ❡q✉❛t✐♦♥s

ax+ by + c = 0,

a1x+ b1y + c1 = 0

}

(∗∗)

✶✼✾

✶✽✵ ❈❍❆P❚❊❘ ✶✻✳ ❚❍❊ ❙❚❘❆■●❍❚ ▲■◆❊

❋✐❣✉r❡ ✶✻✳✶✿ ❊q✉❛t✐♦♥ ♦❢ ❛ ❧✐♥❡

✐s ❝♦♠♣❛t✐❜❧❡✱ ✐t ✐s ❛ ❢♦rt✐♦r✐ s❛t✐s✜❡❞ ❜② t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ♣♦✐♥t B1✱ ❛s✇❡❧❧ ❛s ♦❢ B2✳

❙✐♥❝❡ t❤❡ ♣♦✐♥ts B1✱ ❛♥❞ B2 ❛r❡ ❞✐✛❡r❡♥t✱ t❤❡② ❞✐✛❡r ✐♥ ❛t ❧❡❛st ♦♥❡❝♦♦r❞✐♥❛t❡✱ s❛② y1 6= y2✳ ▼✉❧t✐♣❧②✐♥❣ t❤❡ ✜rst ❡q✉❛t✐♦♥ ♦❢ (∗∗) ❜② a1 ❛♥❞ t❤❡s❡❝♦♥❞ ♦♥❡ ❜② a✱ ❛♥❞ s✉❜tr❛❝t✐♥❣ t❡r♠✇✐s❡✱ ✇❡ ❣❡t

(ba1 − ab1)y + (ca1 − ac1) = 0.

❚❤✐s ❡q✉❛t✐♦♥ ❛s ❛ ❝♦r♦❧❧❛r② ♦❢ t❤❡ ❡q✉❛t✐♦♥s (∗∗) ✐s s❛t✐s✜❡❞ ✇❤❡♥ y = y1❛♥❞ y = y2✳ ❇✉t ✐t ✐s ♣♦ss✐❜❧❡ ♦♥❧② ✐❢

ba1 − ab1 = 0, ca1 − ac1 = 0.

❍❡♥❝❡ ✐t ❢♦❧❧♦✇s t❤❛ta

a1=

b

b1=

c

c1,

✇❤✐❝❤ ♠❡❛♥s t❤❛t t❤❡ ❡q✉❛t✐♦♥s (∗∗) ❛r❡ ❡q✉✐✈❛❧❡♥t✳ ❚❤❡ s❡❝♦♥❞ ♣❛rt ♦❢ t❤❡st❛t❡♠❡♥t ✐s ❛❧s♦ ♣r♦✈❡❞✳

❆s ✇❛s s❤♦✇♥ ✐♥ ❙❡❝t✐♦♥ ✶✺✳✺✱ t❤❡ ♣♦✐♥ts ♦❢ ❛ str❛✐❣❤t ❧✐♥❡ ♣❛ss✐♥❣ t❤r♦✉❣❤(x1, y1) ❛♥❞ (x2, y2) ❛❧❧♦✇ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥

x = (1− t)x1 + tx2, y = (1− t)y1 + ty2.

❲❤❡♥❝❡ ✐t ❢♦❧❧♦✇s t❤❛t ❛♥② str❛✐❣❤t ❧✐♥❡ ❛❧❧♦✇s ❛ ♣❛r❛♠❡tr✐❝ r❡♣r❡s❡♥t❛t✐♦♥❜② ❡q✉❛t✐♦♥s ♦❢ t❤❡ ❢♦r♠

x = at+ b, y = ct+ d, −∞ < t < ∞.

✶✻✳✷✳ P❆❘❚■❈❯▲❆❘ ❈❆❙❊❙ ❖❋ ❚❍❊ ❊◗❯❆❚■❖◆ ❖❋ ❆ ▲■◆❊ ✶✽✶

❋✐❣✉r❡ ✶✻✳✷✿ x✕♣❛r❛❧❧❡❧

❈♦♥✈❡rs❡❧②✱ ❛♥② s✉❝❤ s②st❡♠ ♦❢ ❡q✉❛t✐♦♥s ♠❛② ❜❡ ❝♦♥s✐❞❡r❡❞ ❛s ❡q✉❛t✐♦♥s ♦❢❛ str❛✐❣❤t ❧✐♥❡ ✐♥ ♣❛r❛♠❡tr✐❝ ❢♦r♠ ✐❢ a ❛♥❞ c ❛r❡ ♥♦t ❜♦t❤ ❡q✉❛❧ t♦ ③❡r♦✳ ❚❤✐sstr❛✐❣❤t ❧✐♥❡ ✐s r❡♣r❡s❡♥t❡❞ ❜② t❤❡ ❡q✉❛t✐♦♥ ✐♥ ✐♠♣❧✐❝✐t ❢♦r♠

(x− b)c− (y − d)a = 0.

✶✻✳✷ P❛rt✐❝✉❧❛r ❝❛s❡s ♦❢ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ❧✐♥❡

▲❡t ✉s ✜♥❞ ♦✉t ♣❡❝✉❧✐❛r✐t✐❡s ✇❤✐❝❤ ❤❛♣♣❡♥ ✐♥ t❤❡ ❧♦❝❛t✐♦♥ ♦❢ ❛ str❛✐❣❤t ❧✐♥❡r❡❧❛t✐✈❡ t♦ t❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ✐❢ ✐ts ❡q✉❛t✐♦♥ ax + by + c = 0 ✐s ♦❢ ❛♣❛rt✐❝✉❧❛r ❢♦r♠✳

✶✳ a = 0✳ ■♥ t❤✐s ❝❛s❡ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ str❛✐❣❤t ❧✐♥❡ ❝❛♥ ❜❡ r❡✇r✐tt❡♥ ❛s❢♦❧❧♦✇s

y = −c

b.

❚❤✉s✱ ❛❧❧ ♣♦✐♥ts ❜❡❧♦♥❣✐♥❣ t♦ t❤❡ str❛✐❣❤t ❧✐♥❡ ❤❛✈❡ ♦♥❡ ❛♥❞ t❤❡ s❛♠❡ ♦r❞✐♥❛t❡(−c/b)✱ ❛♥❞✱ ❝♦♥s❡q✉❡♥t❧②✱ t❤❡ ❧✐♥❡ ✐s ♣❛r❛❧❧❡❧ t♦ t❤❡ x✲❛①✐s ✭❋✐❣✳ ✶✻✳✷✮✳ ■♥♣❛rt✐❝✉❧❛r✱ ✐❢ c = 0✱ t❤❡♥ t❤❡ str❛✐❣❤t ❧✐♥❡ ❝♦✐♥❝✐❞❡s ✇✐t❤ t❤❡ x✲❛①✐s✳

✷✳ b = 0✳ ❚❤✐s ❝❛s❡ ✐s ❝♦♥s✐❞❡r❡❞ ✐♥ ❛ s✐♠✐❧❛r ✇❛②✳ ❚❤❡ str❛✐❣❤t ❧✐♥❡ ✐s♣❛r❛❧❧❡❧ t♦ t❤❡ y✲❛①✐s ✭❋✐❣✳ ✶✻✳✸✮ ❛♥❞ ❝♦✐♥❝✐❞❡s ✇✐t❤ ✐t ✐❢ c ✐s ❛❧s♦ ③❡r♦✳

✸✳ c = 0✳ ❚❤❡ str❛✐❣❤t ❧✐♥❡ ♣❛ss❡s t❤r♦✉❣❤ t❤❡ ♦r✐❣✐♥✱ s✐♥❝❡ t❤❡ ❝♦♦r❞✐♥❛t❡s♦❢ t❤❡ ❧❛tt❡r (0, 0) s❛t✐s❢② t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ str❛✐❣❤t ❧✐♥❡ ✭❋✐❣✳ ✶✻✳✹✮✳

✹✳ ❙✉♣♣♦s❡ ❛❧❧ t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ str❛✐❣❤t ❧✐♥❡ ❛r❡♥♦♥✲③❡r♦ ✭✐✳❡✳ t❤❡ ❧✐♥❡ ❞♦❡s ♥♦t ♣❛ss t❤r♦✉❣❤ t❤❡ ♦r✐❣✐♥ ❛♥❞ ✐s ♥♦t ♣❛r❛❧❧❡❧t♦ t❤❡ ❝♦♦r❞✐♥❛t❡ ❛①❡s✮✳ ❚❤❡♥✱ ♠✉❧t✐♣❧②✐♥❣ t❤❡ ❡q✉❛t✐♦♥ ❜② 1/c ❛♥❞ ♣✉tt✐♥❣

✶✽✷ ❈❍❆P❚❊❘ ✶✻✳ ❚❍❊ ❙❚❘❆■●❍❚ ▲■◆❊

❋✐❣✉r❡ ✶✻✳✸✿ y✕♣❛r❛❧❧❡❧

❋✐❣✉r❡ ✶✻✳✹✿ ❆ ❧✐♥❡ t❤r♦✉❣❤ t❤❡ ♦r✐❣✐♥

✶✻✳✸✳ ❊❳❊❘❈■❙❊❙ ✶✽✸

❋✐❣✉r❡ ✶✻✳✺✿ x ❛♥❞ y ✐♥t❡rs❡❝t✐♦♥

−c/a = α✱ −c/b = β✱ ✇❡ r❡❞✉❝❡ ✐t t♦ t❤❡ ❢♦r♠

x

α+

y

β= 1. (∗)

❚❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ str❛✐❣❤t ❧✐♥❡ ✐♥ s✉❝❤ ❛ ❢♦r♠ ✭✇❤✐❝❤ ✐s❝❛❧❧❡❞ t❤❡ ✐♥t❡r❝❡♣t ❢♦r♠ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ str❛✐❣❤t ❧✐♥❡✮ ❤❛✈❡ ❛ s✐♠♣❧❡ ❣❡✲♦♠❡tr✐❝❛❧ ♠❡❛♥✐♥❣✿ α ❛♥❞ β ❛r❡ ❡q✉❛❧ ✭✉♣ t♦ ❛ s✐❣♥✮ t♦ t❤❡ ❧❡♥❣t❤s ♦❢ t❤❡ ❧✐♥❡s❡❣♠❡♥ts ✐♥t❡r❝❡♣t❡❞ ❜② t❤❡ str❛✐❣❤t ❧✐♥❡ ♦♥ t❤❡ ❝♦♦r❞✐♥❛t❡ ❛①❡s ✭❋✐❣✳ ✶✻✳✺✮✳■♥❞❡❡❞✱ t❤❡ str❛✐❣❤t ❧✐♥❡ ✐♥t❡rs❡❝ts ❜♦t❤ t❤❡ x✲❛①✐s (y = 0) ❛t ♣♦✐♥t (α, 0)✱❛♥❞ t❤❡ y✲❛①✐s (x = 0) ❛t ♣♦✐♥t (0, β)✳

✶✻✳✸ ❊①❡r❝✐s❡s

✶✳ ❯♥❞❡r ✇❤❛t ❝♦♥❞✐t✐♦♥ ❞♦❡s t❤❡ str❛✐❣❤t ❧✐♥❡

ax+ by + c = 0

✐♥t❡rs❡❝t t❤❡ ♣♦s✐t✐✈❡ s❡♠✐✲❛①✐s x ✭t❤❡ ♥❡❣❛t✐✈❡ s❡♠✐✲❛①✐s x✮❄✷✳ ❯♥❞❡r ✇❤❛t ❝♦♥❞✐t✐♦♥ ❞♦❡s t❤❡ str❛✐❣❤t ❧✐♥❡

ax+ by + c = 0

♥♦t ✐♥t❡rs❡❝t t❤❡ ✜rst q✉❛❞r❛♥t❄✸✳ ❙❤♦✇ t❤❛t t❤❡ str❛✐❣❤t ❧✐♥❡s ❣✐✈❡♥ ❜② t❤❡ ❡q✉❛t✐♦♥s

ax+ by + c = 0, ax− by + c = 0, b 6= 0,

✶✽✹ ❈❍❆P❚❊❘ ✶✻✳ ❚❍❊ ❙❚❘❆■●❍❚ ▲■◆❊

❛r❡ s✐t✉❛t❡❞ s②♠♠❡tr✐❝❛❧❧② ❛❜♦✉t t❤❡ x✲❛①✐s✳✹✳ ❙❤♦✇ t❤❛t t❤❡ str❛✐❣❤t ❧✐♥❡s s♣❡❝✐✜❡❞ ❜② t❤❡ ❡q✉❛t✐♦♥s

ax+ by + c = 0; ax+ by − c = 0,

❛r❡ ❛rr❛♥❣❡❞ s②♠♠❡tr✐❝❛❧❧② ❛❜♦✉t t❤❡ ♦r✐❣✐♥✳✺✳ ●✐✈❡♥ ❛ ♣❡♥❝✐❧ ♦❢ ❧✐♥❡s

ax+ by + c+ λ(a1x+ b1y + c1) = 0.

❋✐♥❞ ♦✉t ❢♦r ✇❤❛t ✈❛❧✉❡ ♦❢ t❤❡ ♣❛r❛♠❡t❡r λ ✐s ❛ ❧✐♥❡ ♦❢ t❤❡ ♣❡♥❝✐❧ ♣❛r❛❧❧❡❧ t♦t❤❡ x✲❛①✐s ✭y✲❛①✐s✮❀ ❢♦r ✇❤❛t ✈❛❧✉❡ ♦❢ λ ❞♦❡s t❤❡ ❧✐♥❡ ♣❛ss t❤r♦✉❣❤ t❤❡ ♦r✐❣✐♥❄

✻✳ ❯♥❞❡r ✇❤❛t ❝♦♥❞✐t✐♦♥ ❞♦❡s t❤❡ str❛✐❣❤t ❧✐♥❡

ax+ by + c = 0

❜♦✉♥❞✱ t♦❣❡t❤❡r ✇✐t❤ t❤❡ ❝♦♦r❞✐♥❛t❡ ❛①❡s✱ ❛♥ ✐s♦s❝❡❧❡s tr✐❛♥❣❧❡❄✼✳ ❙❤♦✇ t❤❛t t❤❡ ❛r❡❛ ♦❢ t❤❡ tr✐❛♥❣❧❡ ❜♦✉♥❞❡❞ ❜② t❤❡ str❛✐❣❤t ❧✐♥❡

ax+ by + c = 0 (a, b.c 6= 0)

❛♥❞ t❤❡ ❝♦♦r❞✐♥❛t❡ ❛①❡s ✐s

S =1

2

c2

|ab| .

✽✳ ❋✐♥❞ t❤❡ t❛♥❣❡♥t ❧✐♥❡s t♦ t❤❡ ❝✐r❝❧❡

x2 + y2 + 2ax+ 2by = 0,

✇❤✐❝❤ ❛r❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ ❝♦♦r❞✐♥❛t❡ ❛①❡s✳

✶✻✳✹ ❚❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ t✇♦ str❛✐❣❤t ❧✐♥❡s

❲❤❡♥ ♠♦✈✐♥❣ ❛❧♦♥❣ ❛♥② str❛✐❣❤t ❧✐♥❡ ♥♦t ♣❛r❛❧❧❡❧ t♦ t❤❡ y✲❛①✐s x ✐♥❝r❡❛s❡s ✐♥♦♥❡ ❞✐r❡❝t✐♦♥ ❛♥❞ ❞❡❝r❡❛s❡s ✐♥ t❤❡ ♦t❤❡r✳ ❚❤❡ ❞✐r❡❝t✐♦♥ ✐♥ ✇❤✐❝❤ x ✐♥❝r❡❛s❡s✇✐❧❧ ❜❡ ❝❛❧❧❡❞ ♣♦s✐t✐✈❡✳

❙✉♣♣♦s❡ ✇❡ ❛r❡ ❣✐✈❡♥ t✇♦ str❛✐❣❤t ❧✐♥❡s g1 ❛♥❞ g2 ✐♥ t❤❡ xy✲♣❧❛♥❡ ✇❤✐❝❤❛r❡ ♥♦t ♣❛r❛❧❧❡❧ t♦ t❤❡ y✲❛①✐s✳ ❚❤❡ ❛♥❣❧❡ θ(g1, g2) ❢♦r♠❡❞ ❜② t❤❡ ❧✐♥❡ g2 ✇✐t❤t❤❡ ❧✐♥❡ g1 ✐s ❞❡✜♥❡❞ ❛s ❛♥ ❛♥❣❧❡✱ ❧❡ss t❤❛♥ π ❜② ❛❜s♦❧✉t❡ ✈❛❧✉❡✱ t❤r♦✉❣❤ ✇❤✐❝❤t❤❡ ❧✐♥❡ g1 ♠✉st ❜❡ t✉r♥❡❞ s♦ t❤❛t t❤❡ ♣♦s✐t✐✈❡ ❞✐r❡❝t✐♦♥ ♦♥ ✐t ✐s ❜r♦✉❣❤t ✐♥❝♦✐♥❝✐❞❡♥❝❡ ✇✐t❤ t❤❡ ♣♦s✐t✐✈❡ ❞✐r❡❝t✐♦♥ ♦♥ g2✳ ❚❤✐s ❛♥❣❧❡ ✐s ❝♦♥s✐❞❡r❡❞ t♦ ❜❡♣♦s✐t✐✈❡ ✐❢ t❤❡ ❧✐♥❡ g1 ✐s t✉r♥❡❞ ✐♥ t❤❡ s❛♠❡ ❞✐r❡❝t✐♦♥ ✐♥ ✇❤✐❝❤ t❤❡ ♣♦s✐t✐✈❡s❡♠✐✲❛①✐s x ✐s t✉r♥❡❞ t❤r♦✉❣❤ t❤❡ ❛♥❣❧❡ π/2 ✉♥t✐❧ ✐t ❝♦✐♥❝✐❞❡s ✇✐t❤ t❤❡ ♣♦s✐t✐✈❡s❡♠✐✲❛①✐s y ✭❋✐❣✳ ✶✻✳✻✮✳

❚❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ t❤❡ str❛✐❣❤t ❧✐♥❡s ♣♦ss❡ss❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ♦❜✈✐♦✉s ♣r♦♣✲❡rt✐❡s ✿

✶✻✳✹✳ ❚❍❊ ❆◆●▲❊ ❇❊❚❲❊❊◆ ❚❲❖ ❙❚❘❆■●❍❚ ▲■◆❊❙ ✶✽✺

❋✐❣✉r❡ ✶✻✳✻✿ ❆♥❣❧❡ ♦❢ t✇♦ ❧✐♥❡s

✭✶✮ θ(g1, g2) = θ(g2, g1)❀

✭✷✮ θ(g1, g2) = 0 ✇❤❡♥ ❛♥❞ ♦♥❧② ✇❤❡♥ ❧✐♥❡s ❛r❡ ♣❛r❛❧❧❡❧ ♦r ❝♦✐♥❝✐❞❡❀

✭✸✮ θ(g3, g1) = θ(g3, g2) + θ(g2, g1)✳

▲❡tax+ by + c = 0

❜❡ ❛ str❛✐❣❤t ❧✐♥❡ ♥♦t ♣❛r❛❧❧❡❧ t♦ t❤❡ y✲❛①✐s (b 6= 0)✳ ▼✉❧t✐♣❧②✐♥❣ t❤❡ ❡q✉❛t✐♦♥♦❢ t❤❡ ✐s ❧✐♥❡ ❜② 1/b ❛♥❞ ♣✉tt✐♥❣ −a/b = k✱ −c/b = ℓ✱ ✇❡ r❡❞✉❝❡ ✐t t♦ t❤❡❢♦r♠

y = kx+ ℓ. (∗)❚❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ str❛✐❣❤t ❧✐♥❡ ✐♥ t❤✐s ❢♦r♠ ❤❛✈❡ ❛

s✐♠♣❧❡ ❣❡♦♠❡tr✐❝❛❧ ♠❡❛♥✐♥❣✿k ✐s t❤❡ t❛♥❣❡♥t ♦❢ t❤❡ ❛♥❣❧❡ α ❢♦r♠❡❞ ❜② str❛✐❣❤t ❧✐♥❡ ✇✐t❤ t❤❡ x✲❛①✐s ❀ℓ ✐s t❤❡ ❧✐♥❡ s❡❣♠❡♥t ✭✉♣ t♦ ❛ s✐❣♥✮ ✐♥t❡r❝❡♣t❡❞ ❜② t❤❡ str❛✐❣❤t ❧✐♥❡ ♦♥ t❤❡

y✲❛①✐s✳■♥❞❡❡❞✱ ❧❡t A1(x1, y1) ❛♥❞ A2(x2, y2) ❜❡ t✇♦ ♣♦✐♥ts ♦♥ t❤❡ str❛✐❣❤t ❧✐♥❡

✭❋✐❣✳ ✶✻✳✼✮✳ ❚❤❡♥

tanα =y2 − y1x2 − x1

=(kx2 + ℓ)− (kx1 + ℓ)

x2 − x1

= k.

❚❤❡ y✲❛①✐s (x = 0) ✐s ♦❜✈✐♦✉s❧② ✐♥t❡rs❡❝t❡❞ ❜② t❤❡ ❧✐♥❡ ❛t ♣♦✐♥t (0, ℓ)✳▲❡t t❤❡r❡ ❜❡ ❣✐✈❡♥ ✐♥ t❤❡ xy✲♣❧❛♥❡ t✇♦ str❛✐❣❤t ❧✐♥❡s✿

y = k1x+ ℓ1,

y = k2x+ ℓ2.

✶✽✻ ❈❍❆P❚❊❘ ✶✻✳ ❚❍❊ ❙❚❘❆■●❍❚ ▲■◆❊

❋✐❣✉r❡ ✶✻✳✼✿ ❚❤❡ s❧♦♣❡ ♦❢ ❛ ❧✐♥❡

▲❡t ✉s ✜♥❞ t❤❡ ❛♥❣❧❡ θ ❢♦r♠❡❞ ❜② t❤❡ s❡❝♦♥❞ ❧✐♥❡ ✇✐t❤ t❤❡ ✜rst ♦♥❡✳ ❉❡♥♦t✐♥❣❜② α1 ❛♥❞ α2 t❤❡ ❛♥❣❧❡s ❢♦r♠❡❞ ❜② t❤❡ str❛✐❣❤t ❧✐♥❡s ✇✐t❤ t❤❡ x✲❛①✐s✱ ❜② ✈✐rt✉❡♦❢ ♣r♦♣❡rt② ✭✸✮ ✇❡ ❣❡t

θ = α2 − α1.

❙✐♥❝❡ t❤❡ ❛♥❣✉❧❛r ❝♦❡✣❝✐❡♥ts k1 = tanα1✱ k2 = tanα2✱ ✇❡ ❣❡t

tan θ =k2 − k11 + k1k2

.

❲❤❡♥❝❡ θ ✐s ❞❡t❡r♠✐♥❡❞✱ s✐♥❝❡ |θ| < π✳

✶✻✳✺ ❊①❡r❝✐s❡s

✶✳ ❙❤♦✇ t❤❛t t❤❡ str❛✐❣❤t ❧✐♥❡s ax + by + c = 0 ❛♥❞ bx − ay + c′ = 0✐♥t❡rs❡❝t ❛t r✐❣❤t ❛♥❣❧❡s✳

✷✳ ❲❤❛t ❛♥❣❧❡ ✐s ❢♦r♠❡❞ ✇✐t❤ t❤❡ x✲❛①✐s ❜② t❤❡ str❛✐❣❤t ❧✐♥❡

y = x cotα, ✐❢ − π

2< α0?

✸✳ ❋♦r♠ t❤❡ ❡q✉❛t✐♦♥s ♦❢ t❤❡ s✐❞❡s ♦❢ ❛ r✐❣❤t✲❛♥❣❧❡❞ tr✐❛♥❣❧❡ ✇❤♦s❡ s✐❞❡✐s ❡q✉❛❧ t♦ ✶✱ t❛❦✐♥❣ ♦♥❡ ♦❢ t❤❡ s✐❞❡s ❛♥❞ t❤❡ ❛❧t✐t✉❞❡ ❢♦r t❤❡ ❝♦♦r❞✐♥❛t❡ ❛①❡s✳

✹✳ ❋✐♥❞ t❤❡ ✐♥t❡r✐♦r ❛♥❣❧❡s ♦❢ ❤❡ tr✐❛♥❣❧❡ ❜♦✉♥❞❡❞ ❜② t❤❡ str❛✐❣❤t ❧✐♥❡sx+ 2y = 0✱ 2x+ y = 0✱ ❛♥❞ x+ y = 1✳

✺✳ ❯♥❞❡r ✇❤❛t ❝♦♥❞✐t✐♦♥ ❢♦r t❤❡ str❛✐❣❤t ❧✐♥❡s ax+by = 0 ❛♥❞ a1x+b1y =0 ✐s t❤❡ x✲❛①✐s t❤❡ ❜✐s❡❝t♦r ♦❢ t❤❡ ❛♥❣❧❡s ❢♦r♠❡❞ ❜② t❤❡♠❄

✶✻✳✻✳ ❚❍❊ P❆❘❆▲▲❊▲■❙▼ ❆◆❉ P❊❘P❊◆❉■❈❯▲❆❘■❚❨ ❖❋ ▲■◆❊❙ ✶✽✼

✻✳ ❉❡r✐✈❡ t❤❡ ❢♦r♠✉❧❛ tan θ = ca❢♦r t❤❡ ❛♥❣❧❡ θ ❢♦r♠❡❞ ❜② t❤❡ str❛✐❣❤t

❧✐♥❡ x = at+ b✱ y = ct+ d ✇✐t❤ t❤❡ x✲❛①✐s✳✼✳ ❋✐♥❞ t❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ t❤❡ str❛✐❣❤t ❧✐♥❡s r❡♣r❡s❡♥t❡❞ ❜② t❤❡ ❡q✉❛t✐♦♥s

✐♥ ♣❛r❛♠❡tr✐❝ ❢♦r♠✿

x = a1t+ b1,

y = a2t+ b2;

}

x = c1t+ d1,

y = c2t+ d2.

}

✽✳ ❙❤♦✇ t❛❤t t❤❡ q✉❛❞r✐❧❛t❡r❛❧ ❜♦✉♥❞❡❞ ❜② t❤❡ str❛✐❣❤t ❧✐♥❡s

±ax± by + c = 0 (a, b, c 6= 0),

✐s ❛ r❤♦♠❜✉s ❛♥❞ t❤❡ ❝♦♦r❞✐♥❛t❡ ❛①❡s ❛r❡ ✐ts ❞✐❛❣♦♥❛❧s✳

✶✻✳✻ ❚❤❡ ♣❛r❛❧❧❡❧✐s♠ ❛♥❞ ♣❡r♣❡♥❞✐❝✉❧❛r✐t② ♦❢

❧✐♥❡s

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ✐♥ t❤❡ xy✲♣❧❛♥❡ t✇♦ str❛✐❣❤t ❧✐♥❡s ❣✐✈❡♥ ❜② t❤❡ ❡q✉❛t✐♦♥s

a1x+ b1y + c1 = 0,

a2x+ b2y + c2 = 0.

▲❡t ✉s ✜♥❞ ♦✉t ✇❤❛t ❝♦♥❞✐t✐♦♥ ♠✉st ❜❡ s❛t✐s✜❡❞ ❜② t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡❡q✉❛t✐♦♥s ♦❢ t❤❡ str❛✐❣❤t ❧✐♥❡s ❢♦r t❤❡s❡ ❧✐♥❡s ❢♦r t❤❡s❡ ❧✐♥❡s t♦ ❜❡ ✭❛✮ ♣❛r❛❧❧❡❧t♦ ❡❛❝❤ ♦t❤❡r✱ ✭❜✮ ♠✉t✉❛❧❧② ♣❡r♣❡♥❞✐❝✉❧❛r✳

❆ss✉♠❡ t❤❛t ♥❡✐t❤❡r ♦❢ t❤❡ str❛✐❣❤t ❧✐♥❡s ✐s ♣❛r❛❧❧❡❧ t♦ t❤❡ y✲❛①✐s✳ ❚❤❡♥t❤❡✐r ❡q✉❛t✐♦♥s ♠❛② ❜❡ ✇r✐tt❡♥ ✐♥ t❤❡ ❢♦r♠

y = k1x+ ℓ1, y = k2x+ ℓ2,

✇❤❡r❡k1 = −a1

b1, k2 = −a2

b2.

❚❛❦✐♥❣ ✐♥t♦ ❛❝❝♦✉♥t t❤❡ ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ str❛✐❣❤t ❧✐♥❡s✱ ✇❡❣❡t t❤❡ ♣❛r❛❧❧❡❧✐s♠ ❝♦♥❞✐t✐♦♥ ♦❢ t✇♦ str❛✐❣❤t ❧✐♥❡s✿

k1 − k2 = 0,

♦ra1b2 − a2b1 = 0. (∗)

✶✽✽ ❈❍❆P❚❊❘ ✶✻✳ ❚❍❊ ❙❚❘❆■●❍❚ ▲■◆❊

❚❤❡ ♣❡r♣❡♥❞✐❝✉❧❛r✐t② ❝♦♥❞✐t✐♦♥ ♦❢ str❛✐❣❤t ❧✐♥❡s✿

1 + k1k2 = 0,

♦ra1a2 + b1b2 = 0. (∗)

❚❤♦✉❣❤t t❤❡ ❝♦♥❞✐t✐♦♥s (∗) ❛♥❞ (∗∗) ❛r❡ ♦❜t❛✐♥❡❞ ✐♥ t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t♥❡✐t❤❡r ♦❢ t❤❡ str❛✐❣❤t ❧✐♥❡s ✐s ♣❛r❛❧❧❡❧ t♦ t❤❡ y✲❛①✐s✱ t❤❡② r❡♠❛✐♥ tr✉❡ ❡✈❡♥ ✐❢t❤✐s ❝♦♥❞✐t✐♦♥ ✐s ✈✐♦❧❛t❡❞✳

▲❡t ❢♦r ✐♥st❛♥❝❡✱ t❤❡ ✜rst str❛✐❣❤t ❧✐♥❡ ❜❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ y✲❛①✐s✳ ❚❤✐s♠❡❛♥s t❤❛t✱ b1 = 0✳ ■❢ t❤❡ s❡❝♦♥❞ ❧✐♥❡ ✐s ♣❛r❛❧❧❡❧ t♦ t❤❡ ✜rst ♦♥❡✱ t❤❡♥ ✐t ✐s❛❧s♦ ♣❛r❛❧❧❡❧ t♦ t❤❡ y✲❛①✐s✱ ❛♥❞✱ ❝♦♥s❡q✉❡♥t❧②✱ b2 = 0✳ ❚❤❡ ❝♦♥❞✐t✐♦♥ (∗) ✐s♦❜✈✐♦✉s❧② ❢✉❧✜❧❧❡❞✳ ■❢ t❤❡ s❡❝♦♥❞ ❧✐♥❡ ✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ✜rst ♦♥❡✱ t❤❡♥ ✐t✐s ♣❛r❛❧❧❡❧ t♦ t❤❡ x✲❛①✐s ❛♥❞✱ ❝♦♥s❡q✉❡♥t❧②✱ a2 = 0✳ ✐♥ t❤✐s ❝❛s❡ t❤❡ ❝♦♥❞✐t✐♦♥(∗∗) ✐s ♦❜✈✐♦✉s❧② ❢✉❧✜❧❧❡❞✳

▲❡t ✉s ♥♦✇ s❤♦✇ t❤❛t ✐❢ t❤❡ ❝♦♥❞✐t✐♦♥ (∗) ✐s ❢✉❧✜❧❧❡❞ ❢♦r t❤❡ str❛✐❣❤t ❧✐♥❡s✱t❤❡♥ t❤❡② ❛r❡ ❡✐t❤❡r ♣❛r❛❧❧❡❧✱ ♦r ❝♦✐♥❝✐❞❡✳

❙✉♣♣♦s❡✱ b1 6= 0✳ ❚❤❡♥ ✐t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❝♦♥❞✐t✐♦♥ (∗) t❤❛t b2 6= 0✱ s✐♥❝❡✐❢ b2 = 0✱ t❤❡♥ a2 ✐s ❛❧s♦ ❡q✉❛❧ t♦ ③❡r♦ ✇❤✐❝❤ ✐s ✐♠♣♦ss✐❜❧❡✳ ■♥ t❤✐s ❡✈❡♥t t❤❡❝♦♥❞✐t✐♦♥ (∗) ♠❛② ❜❡ ✇r✐tt❡♥ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✇❛②

−a1b1

= −a2b2, ♦r k1 = k2,

✇❤✐❝❤ ❡①♣r❡ss❡s t❤❡ ❡q✉❛❧✐t② ♦❢ t❤❡ ❛♥❣❧❡s ❢♦r♠❡❞ ❜② t❤❡ str❛✐❣❤t ❧✐♥❡s ✇✐t❤t❤❡ x✲❛①✐s✳ ❍❡♥❝❡✱ t❤❡ ❧✐♥❡s ❛r❡ ❡✐t❤❡r ♣❛r❛❧❧❡❧✱ ♦r ❝♦✐♥❝✐❞❡✳

■❢ b1 = 0 ✭✇❤✐❝❤ ♠❡❛♥s t❤❛t a1 6= 0✮✱ t❤❡♥ ✐t ❢♦❧❧♦✇s ❢r♦♠ (∗) t❤❛t b2 = 0✳❚❤✉s✱ ❜♦t❤ str❛✐❣❤t ❧✐♥❡s ❛r❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ y✲❛①✐s ❛♥❞✱ ❝♦♥s❡q✉❡♥t❧②✱ t❤❡②❛r❡ ❡✐t❤❡r ♣❛r❛❧❧❡❧ t♦ ❡❛❝❤ ♦t❤❡r✱ ♦r ❝♦✐♥❝✐❞❡✳

▲❡t ✉s s❤♦✇ t❤❛t t❤❡ ❝♦♥❞✐t✐♦♥ (∗∗) ✐s s✉✣❝✐❡♥t ❢♦r t❤❡ ❧✐♥❡s t♦ ❜❡ ♠✉t✉❛❧❧②♣❡r♣❡♥❞✐❝✉❧❛r✳

❙✉♣♣♦s❡ b1 6= 0 ❛♥❞ b2 6= 0✳ ❚❤❡♥ t❤❡ ❝♦♥❞✐t✐♦♥ (∗∗) ♠❛② ❜❡ r❡✇r✐tt❡♥ ❛s❢♦❧❧♦✇s✿

1 +

(

−a1b1

)(

−a2b2

)

= 0,

♦r1 + k1k2 = 0.

❚❤✐s ♠❡❛♥s t❤❛t t❤❡ str❛✐❣❤t ❧✐♥❡s ❢♦r♠ ❛ r✐❣❤t ❛♥❣❧❡✱ ✐✳❡✳ t❤❡② ❛r❡ ♠✉t✉❛❧❧②♣❡r♣❡♥❞✐❝✉❧❛r✳

■❢ t❤❡♥ b1 = 0 ✭❤❡♥❝❡✱ a0 6= 0✮✱ ✇❡ ❣❡t ❢r♦♠ t❤❡ ❝♦♥❞✐t✐♦♥ (∗∗) t❤❛t a2 = 0✳❚❤✉s✱ t❤❡ ✜rst ❧✐♥❡ ✐s ♣❛r❛❧❧❡❧ t♦ t❤❡ y✲❛①✐s✱ ❛♥❞ t❤❡ s❡❝♦♥❞ ♦♥❡ ✐s ♣❛r❛❧❧❡❧ t♦t❤❡ x✲❛①✐s ✇❤✐❝❤ ♠❡❛♥s t❤❛t t❤❡② ❛r❡ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ ❡❛❝❤ ♦t❤❡r✳

❚❤❡ ❝❛s❡ ✇❤❡♥ b2 = 0 ✐s ❝♦♥s✐❞❡r❡❞ ❛♥❛❧♦❣♦✉s❧②✳

✶✻✳✼✳ ❊❳❊❘❈■❙❊❙ ✶✽✾

✶✻✳✼ ❊①❡r❝✐s❡s

✶✳ ❙❤♦✇ t❤❛t t✇♦ str❛✐❣❤t ❧✐♥❡s ✐♥t❡r❝❡♣t✐♥❣ ♦♥ t❤❡ ❝♦♦r❞✐♥❛t❡ ❛①❡s s❡❣✲♠❡♥ts ♦❢ ❡q✉❛❧ ❧❡♥❣t❤s ❛r❡ ❡✐t❤❡r ♣❛r❛❧❧❡❧✱ ♦r ♣❡r♣❡♥❞✐❝✉❧❛r t♦ ❡❛❝❤ ♦t❤❡r✳

✷✳ ❋✐♥❞ t❤❡ ♣❛r❛❧❧❡❧✐s♠ ✭♣❡r♣❡♥❞✐❝✉❧❛r✐t②✮ ❝♦♥❞✐t✐♦♥ ♦❢ t❤❡ str❛✐❣❤t ❧✐♥❡sr❡♣r❡s❡♥t❡❞ ❜② t❤❡ ❡q✉❛t✐♦♥s ✐♥ ♣❛r❛♠❡tr✐❝ ❢♦r♠✿

x = α1t+ a1,

y = β1t+ b1,

}

x = α2t+ a2,

y = β2t+ b2.

}

✸✳ ❋✐♥❞ t❤❡ ♣❛r❛❧❧❡❧✐s♠ ✭♣❡r♣❡♥❞✐❝✉❧❛r✐t②✮ ❝♦♥❞✐t✐♦♥ ❢♦r t✇♦ str❛✐❣❤t ❧✐♥❡s♦♥❡ ♦❢ ✇❤✐❝❤ ✐s s♣❡❝✐✜❡❞ ❜② t❤❡ ❡q✉❛t✐♦♥

ax+ by + c = 0,

t❤❡ ♦t❤❡r ❜❡✐♥❣ r❡♣r❡s❡♥t❡❞ ♣❛r❛♠❡tr✐❝❛❧❧②✿

x = αt+ β, y = γt+ δ.

✹✳ ■♥ ❛ ❢❛♠✐❧② ♦❢ str❛✐❣❤t ❧✐♥❡s ❣✐✈❡♥ ❜② t❤❡ ❡q✉❛t✐♦♥s

a1x+ b1y + c1 + λ(a2x+ b2y + c2) = 0

✭λ✱ ♣❛r❛♠❡t❡r ♦❢ t❤❡ ❢❛♠✐❧②✮ ✜♥❞ t❤❡ ❧✐♥❡ ♣❛r❛❧❧❡❧ ✭♣❡r♣❡♥❞✐❝✉❧❛r✮ t♦ t❤❡str❛✐❣❤t ❧✐♥❡

ax+ by + c = 0.

✶✻✳✽ ❇❛s✐❝ ♣r♦❜❧❡♠s ♦♥ t❤❡ str❛✐❣❤t ❧✐♥❡

▲❡t ✉s ❢♦r♠ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛♥ ❛r❜✐tr❛r② str❛✐❣❤t ❧✐♥❡ ♣❛ss✐♥❣ t❤r♦✉❣❤ t❤❡♣♦✐♥t A(x1, y1)✳

❙✉♣♣♦s❡ax+ by + c = 0 (∗)

✐s t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ r❡q✉✐r❡❞ ❧✐♥❡✳ ❙✐♥❝❡ t❤❡ ❧✐♥❡ ♣❛ss❡s t❤r♦✉❣❤ t❤❡ ♣♦✐♥tA✱ ✇❡ ❣❡t

ax1 + by1 + c = 0.

❊①♣r❡ss✐♥❣ c ❛♥❞ s✉❜st✐t✉t✐♥❣ ✐t ✐♥ t❤❡ ❡q✉❛t✐♦♥ (∗)✱ ✇❡ ♦❜t❛✐♥

a(x− x1) + b(y − y1) = 0.

✶✾✵ ❈❍❆P❚❊❘ ✶✻✳ ❚❍❊ ❙❚❘❆■●❍❚ ▲■◆❊

■t ✐s ♦❜✈✐♦✉s t❤❛t✱ ❢♦r ❛♥② a ❛♥❞ b✱ t❤❡ str❛✐❣❤t ❧✐♥❡ ❣✐✈❡♥ ❜② t❤✐s ❡q✉❛t✐♦♥♣❛ss❡s t❤r♦✉❣❤ t❤❡ ♣♦✐♥t A✳

▲❡t ✉s ❢♦r♠ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ str❛✐❣❤t ❧✐♥❡ ♣❛ss✐♥❣ t❤r♦✉❣❤ t✇♦ ❣✐✈❡♥♣♦✐♥ts A1(x1, y1)✱ A2(x2, y2)✳

❙✐♥❝❡ t❤❡ str❛✐❣❤t ❧✐♥❡ ♣❛ss❡s t❤r♦✉❣❤ t❤❡ ♣♦✐♥t A1✱ ✐ts ❡q✉❛t✐♦♥ ♠❛② ❜❡✇r✐tt❡♥ ✐♥ t❤❡ ❢♦r♠

a(x− x1) + b(y − y1) = 0.

❙✐♥❝❡ t❤❡ ❧✐♥❡ ♣❛ss❡s t❤r♦✉❣❤ t❤❡ ♣♦✐♥t A2✱ ✇❡ ❤❛✈❡

a(x2 − x1) + b(y2 − y1) = 0,

✇❤❡♥❝❡a

b= − y2 − y1

x2 − x1

,

❛♥❞ t❤❡ r❡q✉✐r❡❞ ❡q✉❛t✐♦♥ ✇✐❧❧ ❜❡

x− x1

x2 − x1

− y − y1y2 − y1

= 0.

▲❡t ✉s ♥♦✇ ❢♦r♠ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ str❛✐❣❤t ❧✐♥❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ ❧✐♥❡

ax+ by + c = 0,

❛♥❞ ♣❛ss✐♥❣ t❤r♦✉❣❤ t❤❡ ♣♦✐♥t A(x1, y1)✳❲❤❛t❡✈❡r t❤❡ ✈❛❧✉❡ ♦❢ λ✱ t❤❡ ❡q✉❛t✐♦♥

ax+ by + λ = 0

r❡♣r❡s❡♥ts ❛ str❛✐❣❤t ❧✐♥❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ ❣✐✈❡♥ ♦♥❡✳ ▲❡t ✉s ❝❤♦♦s❡ λ s♦ t❤❛tt❤❡ ❡q✉❛t✐♦♥ ✐s s❛t✐s✜❡❞ ❢♦r x = x1 ❛♥❞ y = y1✿

ax1 + by1 + λ = 0.

❍❡♥❝❡λ = −ax1 − by,

❛♥❞ t❤❡ r❡q✉✐r❡❞ ❡q✉❛t✐♦♥ ✇✐❧❧ ❜❡

a(x− x1) + b(y − y1) = 0.

▲❡t ✉s ❢♦r♠ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ str❛✐❣❤t ❧✐♥❡ ♣❛ss✐♥❣ t❤r♦✉❣❤ t❤❡ ❣✐✈❡♥ ♣♦✐♥tA(x1, y1) ❛♥❞ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ❧✐♥❡

ax+ by + c = 0.

✶✻✳✽✳ ❇❆❙■❈ P❘❖❇▲❊▼❙ ❖◆ ❚❍❊ ❙❚❘❆■●❍❚ ▲■◆❊ ✶✾✶

❋♦r ❛♥② λ t❤❡ str❛✐❣❤t ❧✐♥❡

bx− ay + λ = 0

✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ❣✐✈❡♥ ❧✐♥❡✳ ❈❤♦♦s✐♥❣ λ s♦ t❤❛t t❤❡ ❡q✉❛t✐♦♥ ✐s s❛t✐s✜❡❞❢♦r x = x1✱ y = y1 ✇❡ ✜♥❞ t❤❡ r❡q✉✐r❡❞ ❡q✉❛t✐♦♥

b(x− x1)− a(y − y1) = 0.

▲❡t ✉s ❢♦r♠ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ str❛✐❣❤t ❧✐♥❡ ♣❛ss✐♥❣ t❤r♦✉❣❤ t❤❡ ❣✐✈❡♥ ♣♦✐♥tA(x1, y1) ❛t ❛♥ ❛♥❣❧❡ α t♦ t❤❡ x✲❛①✐s✳

❚❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ str❛✐❣❤t ❧✐♥❡ ❝❛♥ ❜❡ ✇r✐tt❡♥ ✐♥ t❤❡ ❢♦r♠

y = kx+ ℓ.

❚❤❡ ❝♦❡✣❝✐❡♥ts k ❛♥❞ ℓ ❛r❡ ❢♦✉♥❞ ❢r♦♠ t❤❡ ❝♦♥❞✐t✐♦♥s

tanα = k, y1 = kx1 + ℓ.

❚❤❡ r❡q✉✐r❡❞ ❡q✉❛t✐♦♥ ✐s

y − y1 = (x− x1) tanα.

❲❡ ❝♦♥❝❧✉❞❡ ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛ss❡rt✐♦♥✿ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛♥② str❛✐❣❤t❧✐♥❡ ♣❛ss✐♥❣ t❤r♦✉❣❤ t❤❡ ♣♦✐♥t ♦❢ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t✇♦ ❣✐✈❡♥ str❛✐❣❤t ❧✐♥❡s

a1x+ b1y + c1 = 0, a2x+ b2y + c2 = 0,

❝❛♥ ❜❡ ✇r✐tt❡♥ ✐♥ t❤❡ ❢♦r♠

λ(a1x+ b1y + c1) + µ(a2x+ b2y + c2) = 0. (∗∗)

■♥❞❡❡❞✱ ❢♦r ❛♥② λ ❛♥❞ µ ✇❤✐❝❤ ❛r❡ ♥♦t ❜♦t❤ ③❡r♦✱ t❤❡ ❡q✉❛t✐♦♥ (∗∗) r❡♣r❡✲s❡♥ts ❛ str❛✐❣❤t ❧✐♥❡ ✇❤✐❝❤ ♣❛ss❡s t❤r♦✉❣❤ t❤❡ ♣♦✐♥t ♦❢ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡ t✇♦❣✐✈❡♥ ❧✐♥❡s✱ s✐♥❝❡ ✐ts ❝♦♦r❞✐♥❛t❡s ♦❜✈✐♦✉s❧② s❛t✐s❢② t❤❡ ❡q✉❛t✐♦♥ (∗∗)✳ ❋✉rt❤❡r✱✇❤❛t❡✈❡r t❤❡ ♣♦✐♥t (x1, y1) ✇❤✐❝❤ ✐s ❞✐✛❡r❡♥t ❢r♦♠ t❤❡ ♣♦✐♥t ♦❢ ✐♥t❡rs❡❝t✐♦♥ ♦❢t❤❡ ❣✐✈❡♥ str❛✐❣❤t ❧✐♥❡s✱ t❤❡ ❧✐♥❡ (∗∗) ♣❛ss❡s t❤r♦✉❣❤ t❤❡ ♣♦✐♥t (x1, y1) ✇❤❡♥

λ = a1x1 + b2y2 + c2, −µ = a1x1 + b1y1 + c1.

❈♦♥s❡q✉❡♥t❧②✱ t❤❡ str❛✐❣❤t ❧✐♥❡s r❡♣r❡s❡♥t❡❞ ❜② (∗∗) ❡①❤❛✉st ❛❧❧ t❤❡ ❧✐♥❡s♣❛ss✐♥❣ t❤r♦✉❣❤ t❤❡ ♣♦✐♥t ♦❢ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡ ❣✐✈❡♥ str❛✐❣❤t ❧✐♥❡s✳

✶✾✷ ❈❍❆P❚❊❘ ✶✻✳ ❚❍❊ ❙❚❘❆■●❍❚ ▲■◆❊

✶✻✳✾ ❊①❡r❝✐s❡s

✶✳ ❋♦r♠ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ str❛✐❣❤t ❧✐♥❡ ♣❛r❛❧❧❡❧ ✭♣❡r♣❡♥❞✐❝✉❧❛r✮ t♦ t❤❡str❛✐❣❤t ❧✐♥❡

ax+ by + c = 0,

♣❛ss✐♥❣ t❤r♦✉❣❤ t❤❡ ♣♦✐♥t ♦❢ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡ str❛✐❣❤t ❧✐♥❡s

a1x+ b1y + c1 = 0, a2x+ b2y + c2 = 0.

✷✳ ❯♥❞❡r ✇❤❛t ❝♦♥❞✐t✐♦♥ ❛r❡ t❤❡ ♣♦✐♥ts (x1, y1)✱ (x2, y2) s✐t✉❛t❡❞ s②♠✲♠❡tr✐❝❛❧❧② ❛❜♦✉t t❤❡ str❛✐❣❤t ❧✐♥❡

ax+ by + c = 0?

✸✳ ❋♦r♠ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ str❛✐❣❤t ❧✐♥❡ ♣❛ss✐♥❣ t❤r♦✉❣❤ t❤❡ ♣♦✐♥t (x0, y0)❛♥❞ ❡q✉✐❞✐st❛♥t ❢r♦♠ t❤❡ ♣♦✐♥ts (x1, y1) ❛♥❞ (x2, y2)✳

✹✳ ❙❤♦✇ t❤❛t t❤r❡❡ ♣♦✐♥ts (x1, y1)✱ (x2, y2) ❛♥❞ (x3, y3) ❧✐❡ ♦♥ ❛ str❛✐❣❤t❧✐♥❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢

∣

∣

∣

∣

∣

∣

x1 y1 1x2 y2 1x3 y3 1

∣

∣

∣

∣

∣

∣

= 0.

❈❤❛♣t❡r ✶✼

❱❡❝t♦rs

✶✼✳✶ ❆❞❞✐t✐♦♥ ❛♥❞ s✉❜tr❛❝t✐♦♥ ♦❢ ✈❡❝t♦rs

■♥ ❣❡♦♠❡tr②✱ ❛ ✈❡❝t♦r ✐s ✉♥❞❡rst♦♦❞ ❛s ❛ ❞✐r❡❝t❡❞ ❧✐♥❡ s❡❣♠❡♥t ✭❋✐❣✳ ✶✼✳✶✮✳❚❤❡ ❞✐r❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✐s ✐♥❞✐❝❛t❡❞ ❜② t❤❡ ❛rr♦✇✳ ❆ ✈❡❝t♦r ✇✐t❤ ✐♥✐t✐❛❧ ♣♦✐♥t

A ❛♥❞ t❡r♠✐♥❛❧ ♣♦✐♥t B ✐s ❞❡♥♦t❡❞ ❛s−→AB✳ ❆ ✈❡❝t♦r ❝❛♥ ❛❧s♦ ❜❡ ❞❡♥♦t❡❞ ❜②

❛ s✐♥❣❧❡ ❧❡tt❡r✳ ■♥ ♣r✐♥t✐♥❣ t❤✐s ❧❡tt❡r ✐s ❣✐✈❡♥ ✐♥ ❜♦❧❞❢❛❝❡ t②♣❡ (a)✱ ✐♥ ✇r✐t✐♥❣✐t ✐s ❣✐✈❡♥ ✇✐t❤ ❛ ❜❛r (a)✳

❚✇♦ ✈❡❝t♦rs ❛r❡ ❝♦♥s✐❞❡r❡❞ t♦ ❜❡ ❡q✉❛❧ ✐❢ ♦♥❡ ♦❢ t❤❡♠ ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞❢r♦♠ t❤❡ ♦t❤❡r ❜② tr❛♥s❧❛t✐♦♥ ✭❋✐❣✳ ✶✼✳✶✮✳ ❖❜✈✐♦✉s❧②✱ ✐❢ t❤❡ ✈❡❝t♦r a ✐s ❡q✉❛❧t♦ b ✐s ❡q✉❛❧ t♦ a✳ ■❢ a ✐s ❡q✉❛❧ t♦ b✱ ❛♥❞ b ✐s ❡q✉❛❧ t♦ c✱ t❤❡♥ a ✐s ❡q✉❛❧ t♦c✳

❚❤❡ ✈❡❝t♦rs ❛r❡ s❛✐❞ t♦ ❜❡ ✐♥ t❤❡ s❛♠❡ ❞✐r❡❝t✐♦♥ ✭✐♥ ♦♣♣♦s✐t❡ ❞✐r❡❝t✐♦♥s✮■❢ t❤❡② ❛r❡ ♣❛r❛❧❧❡❧✱ ❛♥❞ t❤❡ t❡r♠✐♥❛❧ ♣♦✐♥ts ♦❢ t✇♦ ✈❡❝t♦rs ❡q✉❛❧ t♦ t❤❡♠ ❛♥❞r❡❞✉❝❡❞ t♦ ❛ ❝♦♠♠♦♥ ♦r✐❣✐♥ ❛r❡ ❢♦✉♥❞ ♦♥ ♦♥❡ s✐❞❡ ♦❢ t❤❡ ♦r✐❣✐♥ ✭♦♥ ❞✐✛❡r❡♥ts✐❞❡s ♦❢ t❤❡ ♦r✐❣✐♥✮✳

❚❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❧✐♥❡ s❡❣♠❡♥t ❞❡♣✐❝t✐♥❣ ❛ ✈❡❝t♦r ✐s ❝❛❧❧❡❞ t❤❡ ❛❜s♦❧✉t❡✈❛❧✉❡ ♦❢ t❤❡ ✈❡❝t♦r✳

❋✐❣✉r❡ ✶✼✳✶✿ ❱❡❝t♦r r❡♣r❡s❡♥t❛t✐♦♥

✶✾✸

✶✾✹ ❈❍❆P❚❊❘ ✶✼✳ ❱❊❈❚❖❘❙

❋✐❣✉r❡ ✶✼✳✷✿ ❱❡❝t♦r ❛❞❞✐t✐♦♥

❆ ✈❡❝t♦r ♦❢ ③❡r♦ ❧❡♥❣t❤ ✭✐✳❡✳ ✇❤♦s❡ ✐♥✐t✐❛❧ ♣♦✐♥t ❝♦✐♥❝✐❞❡s ✇✐t❤ t❤❡ t❡r♠✐✲♥✉s✮ ✐s t❡r♠❡❞ t❤❡ ③❡r♦ ✈❡❝t♦r✳

❱❡❝t♦rs ♠❛② ❜❡ ❛❞❞❡❞ ♦r s✉❜tr❛❝t❡❞ ❣❡♦♠❡tr✐❝❛❧❧②✱ ✐✳❡✳ ✇❡ ♠❛② s♣❡❛❦ ♦❢❛❞❞✐t✐♦♥ ❛♥❞ s✉❜tr❛❝t✐♦♥ ♦❢ ✈❡❝t♦rs✳ ◆❛♠❡❧②✱ t❤❡ s✉♠ ♦❢ t✇♦ ✈❡❝t♦rs a ❛♥❞ b

✐s ❛ t❤✐r❞ ✈❡❝t♦r a+b ✇❤✐❝❤ ✐s ♦❜t❛✐♥❡❞ ❢r♦♠ t❤❡ ✈❡❝t♦rs a ❛♥❞ b ✭♦r ✈❡❝t♦rs❡q✉❛❧ t♦ t❤❡♠✮ ✐♥ t❤❡ ✇❛② s❤♦✇♥ ✐♥ ❋✐❣✳ ✶✼✳✷✳

❋✐❣✉r❡ ✶✼✳✸✿ ❈♦♠♠✉t❛t✐✈✐t② ♦❢ ✈❡❝t♦r ❛❞❞✐t✐♦♥

❱❡❝t♦r ❛❞❞✐t✐♦♥ ✐s ❝♦♠♠✉t❛t✐✈❡✱ ✐✳❡✳ ❢♦r ❛♥② ✈❡❝t♦rs a ❛♥❞ b ✭❋✐❣✳ ✶✼✳✸✮✳

a+ b = b+ a.

❱❡❝t♦r ❛❞❞✐t✐♦♥ ✐s ❛ss♦❝✐❛t✐✈❡✱ ✐✳❡✳ ✐❢ a✱ b✱ c ❛r❡ ❛♥② ✈❡❝t♦rs t❤❡♥

(a+ b) + c = a+ (b+ c).

❚❤✐s ♣r♦♣❡rt② ♦❢ ❛❞❞✐t✐♦♥✱ ❛s ❛❧s♦ t❤❡ ♣r❡❝❡❞✐♥❣ ♦♥❡✱ ❢♦❧❧♦✇s ❞✐r❡❝t❧② ❢r♦♠t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ♦♣❡r❛t✐♦♥ ♦❢ ❛❞❞✐t✐♦♥ ✭❋✐❣✳ ✶✼✳✹✮✳

▲❡t ✉s ♠❡♥t✐♦♥ ❤❡r❡ t❤❛t ✐❢ t❤❡ ✈❡❝t♦rs a ❛♥❞ b ❛r❡ ♣❛r❛❧❧❡❧✱ t❤❡♥ t❤❡✈❡❝t♦r a + b ✭✐❢ ✐t ✐s ♥♦t ❡q✉❛❧ t♦ ③❡r♦✮ ✐s ♣❛r❛❧❧❡❧ t♦ t❤❡ ✈❡❝t♦rs a ❛♥❞ b✱❛♥❞ ✐s ✐♥ t❤❡ s❛♠❡ ❞✐r❡❝t✐♦♥ ✇✐t❤ t❤❡ ❣r❡❛t❡r ✭❜② ❛❜s♦❧✉t❡ ✈❛❧✉❡✮ ✈❡❝t♦r✳ ❚❤❡❛❜s♦❧✉t❡ ✈❛❧✉❡ ♦❢ t❤❡ ✈❡❝t♦r a+ b ✐s ❡q✉❛❧ t♦ t❤❡ s✉♠ ♦❢ t❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡s♦❢ t❤❡ ✈❡❝t♦rs a ❛♥❞ b ✐❢ t❤❡② ❛r❡ ✐♥ t❤❡ s❛♠❡ ❞✐r❡❝t✐♦♥✱ ❛♥❞ t♦ t❤❡ ❞✐✛❡r❡♥❝❡♦❢ t❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡s ✐❢ t❤❡ ✈❡❝t♦rs a ❛♥❞ b ❛r❡ ✐♥ ♦♣♣♦s✐t❡ ❞✐r❡❝t✐♦♥s✳

✶✼✳✷✳ ❊❳❊❘❈■❙❊❙ ✶✾✺

❋✐❣✉r❡ ✶✼✳✹✿ ❆ss♦❝✐❛t✐✈✐t② ♦❢ ✈❡❝t♦r ❛❞❞✐t✐♦♥

❋✐❣✉r❡ ✶✼✳✺✿ ❱❡❝t♦r s✉❜tr❛❝t✐♦♥

❙✉❜tr❛❝t✐♦♥ ♦❢ ✈❡❝t♦rs ✐s ❞❡✜♥❡❞ ❛s t❤❡ ✐♥✈❡rs❡ ♦♣❡r❛t✐♦♥ ♦❢ ❛❞❞✐t✐♦♥✳◆❛♠❡❧②✱ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ✈❡❝t♦rs a ❛♥❞ b ✐s ❞❡✜♥❡❞ ❛s t❤❡ ✈❡❝t♦r a− b

✇❤✐❝❤✱ t♦❣❡t❤❡r ✇✐t❤ t❤❡ ✈❡❝t♦r b✱ ②✐❡❧❞s t❤❡ ✈❡❝t♦r a✳ ●❡♦♠❡tr✐❝❛❧❧② ✐t ✐s♦❜t❛✐♥❡❞ ❢r♦♠ t❤❡ ✈❡❝t♦rs a ❛♥❞ b ✭♦r ✈❡❝t♦rs ❡q✉❛❧ t♦ t❤❡♠✮ ❛s ✐s s❤♦✇♥ ✐♥❋✐❣✳ ✶✼✳✺✳

❋♦r ❛♥② ✈❡❝t♦rs a ❛♥❞ b ✇❡ ❤❛✈❡ ❢♦❧❧♦✇✐♥❣ ✐♥❡q✉❛❧✐t②

|a+ b| ≤ |a|+ |b|

✭t❤❡ tr✐❛♥❣❧❡ ✐♥❡q✉❛❧✐t②✮✱ ❣❡♦♠❡tr②❝❛❧❧② ❡①♣r❡ss✐♥❣ t❤❡ ❢❛❝t t❤❛t ✐♥ ❛ tr✐❛♥❣❧❡t❤❡ s✉♠ ♦❢ ✐ts t✇♦ s✐❞❡s ✐s ❣r❡❛t❡r t❤❛♥ t❤❡ t❤✐r❞ s✐❞❡ ✐❢ t❤❡ ✈❡❝t♦rs ❛r❡ ♥♦t♣❛r❛❧❧❡❧✳ ❚❤✐s ✐♥❡q✉❛❧✐t② ✐s ♦❜✈✐♦✉s❧② ✈❛❧✐❞ ❢♦r ❛♥② ♥✉♠❜❡r ♦❢ ✈❡❝t♦rs✿

|a+ b+ · · ·+ l| ≤ |a|+ |b|+ · · ·+ |l|.

✶✼✳✷ ❊①❡r❝✐s❡s

✶✳ ❙❤♦✇ t❤❛t t❤❡ s✉♠ ♦❢ n ✈❡❝t♦rs r❡❞✉❝❡❞ t♦ ❛ ❝♦♠♠♦♥ ♦r✐❣✐♥ ❛t t❤❡❝❡♥tr❡ ♦❢ ❛ r❡❣✉❧❛r n✲❣♦♥ ❛♥❞ ✇✐t❤ t❤❡ t❡r♠✐♥❛❧ ♣♦✐♥ts ❛t ✐ts ✈❡rt✐❝❡s ✐s ❡q✉❛❧t♦ ③❡r♦✳

✶✾✻ ❈❍❆P❚❊❘ ✶✼✳ ❱❊❈❚❖❘❙

✷✳ ❚❤r❡❡ ✈❡❝t♦rs ❤❛✈❡ ❛ ❝♦♠♠♦♥ ♦r✐❣✐♥ O ❛♥❞ t❤❡✐r t❡r♠✐♥❛❧ ♣♦✐♥ts ❛r❡❛t t❤❡ ✈❡rt✐❝❡s ♦❢ t❤❡ tr✐❛♥❣❧❡ ABC✳ ❙❤♦✇ t❤❛t

−→OA+

−−→OB +

−→OC = 0

✐❢ ❛♥❞ ♦♥❧② ✐❢ O ✐s t❤❡ ♣♦✐♥t ♦❢ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡ ♠❡❞✐❛♥s ♦❢ t❤❡ tr✐❛♥❣❧❡✳✸✳ Pr♦✈❡ t❤❡ ✐❞❡♥t✐t②

2|a|2 + 2|b|2 = |a+ b|2 + |a− b|2.

❚♦ ✇❤❛t ❣❡♦♠❡tr✐❝❛❧ ❢❛❝t ❞♦❡s ✐t ❝♦rr❡s♣♦♥❞ ✐❢ a ❛♥❞ b ❛r❡ ♥♦♥✲③❡r♦ ❛♥❞♥♦♥✲♣❛r❛❧❧❡❧ ✈❡❝t♦rs❄

✹✳ ❙❤♦✇ t❤❛t t❤❡ s✐❣♥ ♦❢ ❡q✉❛❧✐t② ✐♥ t❤❡ tr✐❛♥❣❧❡ ✐♥❡q✉❛❧✐t② t❛❦❡s ♣❧❛❝❡♦♥❧② ✇❤❡♥ ❜♦t❤ ✈❡❝t♦rs ❛r❡ ✐♥ t❤❡ s❛♠❡ ❞✐r❡❝t✐♦♥✱ ♦r ❛t ❧❡❛st ♦♥❡ ♦❢ t❤❡✈❡❝t♦rs ✐s ❡q✉❛❧ t♦ ③❡r♦✳

✺✳ ■❢ t❤❡ s✉♠ ♦❢ t❤❡ ✈❡❝t♦rs r1, . . . , rn r❡❞✉❝❡❞ t♦ ❛ ❝♦♠♠♦♥ ♦r✐❣✐♥ O ✐s❡q✉❛❧ t♦ ③❡r♦ ❛♥❞ t❤❡s❡ ✈❡❝t♦rs ❛r❡ ♥♦t ❝♦♣❧❛♥❛r✱ t❤❡♥ ✇❤❛t❡✈❡r ✐s t❤❡ ♣❧❛♥❡α ♣❛ss✐♥❣ t❤r♦✉❣❤ t❤❡ ♣♦✐♥t O t❤❡r❡ ❝❛♥ ❜❡ ❢♦✉♥❞ ✈❡❝t♦rs ri s✐t✉❛t❡❞ ♦♥ ❜♦t❤s✐❞❡s ♦❢ t❤❡ ♣❧❛♥❡✳ ❙❤♦✇ t❤✐s✳

✻✳ ❚❤❡ ✈❡❝t♦r rmn ❧✐❡s ✐♥ t❤❡ xy✲♣❧❛♥❡❀ ✐ts ✐♥✐t✐❛❧ ♣♦✐♥t ✐s (x0, y0) ❛♥❞t❤❡ t❡r♠✐♥✉s ✐s t❤❡ ♣♦✐♥t (mδ, nδ)✱ ✇❤❡r❡ m ❛♥❞ n ❛r❡ ✇❤♦❧❡ ♥✉♠❜❡rs ♥♦t❡①❝❡❡❞✐♥❣ M ❛♥❞ N ❜② ❛❜s♦❧✉t❡ ✈❛❧✉❡✱ r❡s♣❡❝t✐✈❡❧②✳ ❋✐♥❞ t❤❡ s✉♠ ♦❢ ❛❧❧ t❤❡✈❡❝t♦rs rmn ❡①♣r❡ss✐♥❣ ✐t ✐♥ t❡r♠s ♦❢ t❤❡ ✈❡❝t♦r ✇✐t❤ t❤❡ ✐♥✐t✐❛❧ ♣♦✐♥t ❛t (0, 0)❛♥❞ t❤❡ t❡r♠✐♥✉s ❛t t❤❡ ♣♦✐♥t (x0, y0)✳

✼✳ ❆ ✜♥✐t❡ ✜❣✉r❡ F ✐♥ t❤❡ xy✲♣❧❛♥❡ ❤❛s t❤❡ ♦r✐❣✐♥ ❛s t❤❡ ❝❡♥tr❡ ♦❢ s②♠✲♠❡tr②✳ ❙❤♦✇ t❤❛t t❤❡ s✉♠ ♦❢ t❤❡ ✈❡❝t♦rs ✇✐t❤ ❛ ❝♦♠♠♦♥ ♦r✐❣✐♥ ❛♥❞ t❡r♠✐♥✐❛t t❤❡ ♣♦✐♥ts ✇❤♦s❡ ❝♦♦r❞✐♥❛t❡s ❛r❡ ✇❤♦❧❡ ♥✉♠❜❡rs ♦❢ t❤❡ ✜❣✉r❡ F ✐s ❡q✉❛❧t♦ ③❡r♦ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ♦r✐❣✐♥ ♦❢ ❝♦♦r❞✐♥❛t❡s s❡r✈❡s ❛s t❤❡✐r ❝♦♠♠♦♥ ✐♥✐✲t✐❛❧ ♣♦✐♥t✳ ✭■t ✐s ❛ss✉♠❡❞ t❤❛t t❤❡ ✜❣✉r❡ F ❤❛s ❛t ❧❡❛st ♦♥❡ ♣♦✐♥t ✇❤♦s❡❝♦♦r❞✐♥❛t❡s ❛r❡ ✇❤♦❧❡ ♥✉♠❜❡rs✳✮

✽✳ ❊①♣r❡ss t❤❡ ✈❡❝t♦rs r❡♣r❡s❡♥t❡❞ ❜② t❤❡ ❞✐❛❣♦♥❛❧s ♦❢ ❛ ♣❛r❛❧❧❡❧❡♣✐♣❡❞✐♥ t❡r♠s ♦❢ ✐♥ t❡r♠s ♦❢ t❤❡ ✈❡❝t♦rs r❡♣r❡s❡♥t❡❞ ❜② ✐ts ❡❞❣❡s✳

✶✼✳✸ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ❜② ❛ ♥✉♠❜❡r

❱❡❝t♦rs ♠❛② ❛❧s♦ ❜❡ ♠✉❧t✐♣❧✐❡❞ ❜② ❛ ♥✉♠❜❡r✳ ❚❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ ✈❡❝t♦r a ❜②t❤❡ ♥✉♠❜❡r λ ✐s ❞❡✜♥❡❞ ❛s t❤❡ ✈❡❝t♦r aλ = λa t❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡ ♦❢ ✇❤✐❝❤✐s ♦❜t❛✐♥❡❞ ❜② ♠✉❧t✐♣❧②✐♥❣ t❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡ ♦❢ t❤❡ ✈❡❝t♦r a ❜② t❤❡ ❛❜s♦❧✉t❡✈❛❧✉❡ ♦❢ t❤❡ ♥✉♠❜❡r λ✱ ✐✳❡✳ |λa| = |λ| |a|✱ t❤❡ ❞✐r❡❝t✐♦♥ ❝♦✐♥❝✐❞✐♥❣ ✇✐t❤ t❤❡❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ✈❡❝t♦r a ♦r ❜❡✐♥❣ ✐♥ t❤❡ ♦♣♣♦s✐t❡ s❡♥s❡ ❞❡♣❡♥❞✐♥❣ ♦♥ ✇❤❡t❤❡rλ > 0 ♦r λ < 0✳ ■❢ λ = 0 ♦r a = 0✱ t❤❡♥ λa ✐s ❝♦♥s✐❞❡r❡❞ t♦ ❜❡ ❡q✉❛❧ t♦ ③❡r♦✈❡❝t♦r✳

✶✼✳✸✳ ▼❯▲❚■P▲■❈❆❚■❖◆ ❖❋ ❆ ❱❊❈❚❖❘ ❇❨ ❆ ◆❯▼❇❊❘ ✶✾✼

❚❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ❜② ❛ ♥✉♠❜❡r ♣♦ss❡ss❡s t❤❡ ❛ss♦❝✐❛t✐✈❡ ♣r♦♣✲❡rt② ❛♥❞ t✇♦ ❞✐str✐❜✉t✐✈❡ ♣r♦♣❡rt✐❡s✳ ◆❛♠❡❧②✱ ❢♦r ❛♥② ♥✉♠❜❡r λ✱ µ ❛♥❞ ✈❡❝t♦rsa✱ b

λ(µa) = (λµ)a ✭❛ss♦❝✐❛t✐✈❡ ♣r♦♣❡rt②✮

(λ+ µ)a = λa+ µa,

λ(a+ b) = λa+ λb

}

✭❞✐str✐❜✉t✐✈❡ ♣r♦♣❡rt✐❡s✮

▲❡t ✉s ♣r♦✈❡ t❤❡s❡ ♣r♦♣❡rt✐❡s✳❚❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡s ♦❢ t❤❡ ✈❡❝t♦rs λ(µa) ❛♥❞ (λµ)a ❛r❡ t❤❡ s❛♠❡ ❛♥❞ ❛r❡

❡q✉❛❧ t♦ |λ| |µ| |a|✳ ❚❤❡ ❞✐r❡❝t✐♦♥s ♦❢ t❤❡s❡ ✈❡❝t♦rs ❡✐t❤❡r ❝♦✐♥❝✐❞❡✱ ✐❢ λ ❛♥❞µ ❛r❡ ♦❢ t❤❡ s❛♠❡ s✐❣♥✱ ♦r ♦♣♣♦s✐t❡ ✐❢ λ ❛♥❞ µ ❤❛✈❡ ❞✐✛❡r❡♥t s✐❣♥s✳ ❍❡♥❝❡✱t❤❡ ✈❡❝t♦rs λ(µa) ❛♥❞ (λµ)a ❛r❡ ❡q✉❛❧ ❜② ❛❜s♦❧✉t❡ ✈❛❧✉❡ ❛♥❞ ❛r❡ ✐♥ ❤❡ s❛♠❡❞✐r❡❝t✐♦♥✱ ❝♦♥s❡q✉❡♥t❧②✱ t❤❡② ❛r❡ ❡q✉❛❧✳ ■❢ ❛t ❧❡❛st ♦♥❡ ♦❢ t❤❡ ♥✉♠❜❡rs λ✱ µ ♦rt❤❡ ✈❡❝t♦r a ✐s ❡q✉❛❧ t♦ ③❡r♦✱ t❤❡♥ ❜♦t❤ ✈❡❝t♦rs ❛r❡ ❡q✉❛❧ t♦ ③❡r♦ ❛♥❞✱ ❤❡♥❝❡✱t❤❡② ❛r❡ ❡q✉❛❧ t♦ ❡❛❝❤ ♦t❤❡r✳ ❚❤❡ ❛ss♦❝✐❛t✐✈❡ ♣r♦♣❡rt② ✐s t❤✉s ♣r♦✈❡❞✳

❲❡ ❛r❡ ♥♦✇ ❣♦✐♥❣ t♦ ♣r♦✈❡ t❤❡ ✜rst ❞✐str✐❜✉t✐✈❡ ♣r♦♣❡rt②✿

(λ+ µ)a = λa+ µa.

❚❤❡ ❡q✉❛❧✐t② ✐s ♦❜✈✐♦✉s ✐❢ ❛t ❧❡❛st ♦♥❡ ♦❢ t❤❡ ♥✉♠❜❡rs λ✱ µ ♦r t❤❡ ✈❡❝t♦r a ✐s❡q✉❛❧ t♦ ③❡r♦✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ♠❛② ❝♦♥s✐❞❡r t❤❛t λ✱ µ✱ ❛♥❞ a ❛r❡ ♥♦♥✲③❡r♦✳

■❢ λ ❛♥❞ µ ❛r❡ ♦❢ t❤❡ s❛♠❡ s✐❣♥✱ t❤❡♥ t❤❡ ✈❡❝t♦rs λa ❛♥❞ µa ❛r❡ ✐♥ t❤❡s❛♠❡ ❞✐r❡❝t✐♦♥✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡ ♦❢ t❤❡ ✈❡❝t♦r λa+ µa ✐s ❡q✉❛❧t♦ |λa| + |µa| = |λ| |a| + |µ| |a| = (|λ| + |µ|)|a|✳ ❚❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡ ♦❢ t❤❡✈❡❝t♦r (λ + µ)a ✐s ❡q✉❛❧ t♦ |λ + µ| |a| = (|λ| + |µ|)|a|✳ ❚❤✉s✱ t❤❡ ❛❜s♦❧✉t❡✈❛❧✉❡s ♦❢ t❤❡ ✈❡❝t♦rs (λ + µ)a ❛♥❞ λa + µa ❛r❡ ❡q✉❛❧ ❛♥❞ t❤❡② ❛r❡ ✐♥ t❤❡s❛♠❡ ❞✐r❡❝t✐♦♥✳ ◆❛♠❡❧②✱ ❢♦r λ > 0✱ µ > 0 t❤❡✐r ❞✐r❡❝t✐♦♥s ❝♦✐♥❝✐❞❡ ✇✐t❤ t❤❡❞✐r❡❝t✐♦♥ ♦❢ a✱ ❛♥❞ ✐❢ λ < 0✱ µ < 0 t❤❡② ❛r❡ ♦♣♣♦s✐t❡ t♦ a✳ ❚❤❡ ❝❛s❡ ✇❤❡♥ λ❛♥❞ µ ❤❛✈❡ ❞✐✛❡r❡♥t s✐❣♥s ✐s ❝♦♥s✐❞❡r❡❞ ✐♥ ❛ s✐♠✐❧❛r ✇❛②✳

❋✐❣✉r❡ ✶✼✳✻✿ ❉✐str✐❜✉t✐✈❡ ❧❛✇

✶✾✽ ❈❍❆P❚❊❘ ✶✼✳ ❱❊❈❚❖❘❙

▲❡t ✉s ♣r♦✈❡ t❤❡ s❡❝♦♥❞ ❞✐str✐❜✉t✐✈❡ ♣r♦♣❡rt②✿

λ(a+ b) = λa+ λb.

❚❤❡ ♣r♦♣❡rt② ✐s ♦❜✈✐♦✉s ✐❢ ♦♥❡ ♦❢ t❤❡ ✈❡❝t♦rs ♦r t❤❡ ♥✉♠❜❡r λ ✐s ❡q✉❛❧ t♦③❡r♦✳ ■❢ t❤❡ ✈❡❝t♦rs a ❛♥❞ b ❛r❡ ♣❛r❛❧❧❡❧✱ t❤❡♥ b ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ✐♥ t❤❡❢♦r♠ b = µa✳ ❆♥❞ t❤❡ s❡❝♦♥❞ ❞✐str✐❜✉t✐✈❡ ♣r♦♣❡rt② ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ✜rst♦♥❡✳ ■♥❞❡❡❞✱

λ(1+ µ)a = λ(a+ µa) = λa+ λµa.

❍❡♥❝❡λ(a+ b) = λa+ λb.

▲❡t a ❛♥❞ b ❜❡ ♥♦♥✲♣❛r❛❧❧❡❧ ✈❡❝t♦rs✱ t❤❡♥ ❢♦r λ > 0 t❤❡ ✈❡❝t♦r−→AB

✭❋✐❣✳ ✶✼✳✻✮ r❡♣r❡s❡♥ts✱ ♦♥ t❤❡ ♦♥❡ ❤❛♥❞✱ λa+λb✱ ❛♥❞ λ−→AC ❡q✉❛❧ t♦ λ(a+b)

♦♥ t❤❡ ♦t❤❡r✳ ■❢ λ < 0✱ t❤❡♥ ❜♦t❤ ✈❡❝t♦rs r❡✈❡rs❡ t❤❡✐r ❞✐r❡❝t✐♦♥s✳

✶✼✳✹ ❊①❡r❝✐s❡s

✶✳ ❚❤❡ ✈❡❝t♦rs r1, r2, . . . ❛r❡ ❝❛❧❧❡❞ ❧✐♥❡❛r❧② ✐♥❞❡♣❡♥❞❡♥t ✐❢ t❤❡r❡ ❡①✐st ♥♦♥✉♠❜❡rs λ1, λ2, . . . ✱ ✭❛t ❧❡❛st ♦♥❡ ♦❢ ✇❤✐❝❤ ✐s ♥♦♥✲③❡r♦✮ s✉❝❤ t❤❛t

λ1r1 + λ2r2 + · · · = 0

❙❤♦✇ t❤❛t t✇♦ ✈❡❝t♦rs ❛r❡ ❧✐♥❡❛r❧② ✐♥❞❡♣❡♥❞❡♥t ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡② ❛r❡♥♦♥✲③❡r♦ ❛♥❞ ♥♦♥✲♣❛r❛❧❧❡❧✳

❙❤♦✇ t❤❛t t❤r❡❡ ✈❡❝t♦rs ❛r❡ ❧✐♥❡❛r❧② ✐♥❞❡♣❡♥❞❡♥t ✇❤❡♥ ❛♥❞ ♦♥❧② ✇❤❡♥t❤❡② ❛r❡ ♥♦♥✲③❡r♦ ❛♥❞ t❤❡r❡ ✐s ♥♦ ♣❧❛♥❡ ♣❛r❛❧❧❡❧ t♦ t❤❡♠✳

✷✳ ❙❤♦✇ t❤❛t ❛♥② t❤r❡❡ ✈❡❝t♦rs ❧②✐♥❣ ✐♥ ♦♥❡ ♣❧❛♥❡ ❛r❡ ❛❧✇❛②s ❧✐♥❡❛r❧②❞❡♣❡♥❞❡♥t✳

✸✳ ❙❤♦✇ t❤❛t ✐❢ t✇♦ ✈❡❝t♦rs r1 ❛♥❞ r2 ✐♥ ❛ ♣❧❛♥❡ ❛r❡ ❧✐♥❡❛r❧② ✐♥❞❡♣❡♥❞❡♥t✱t❤❡♥ ❛♥② ✈❡❝t♦r r ✐♥ t❤✐s ♣❧❛♥❡ ✐s ❡①♣r❡ss❡❞ ❧✐♥❡❛r❧② ✐♥ t❡r♠s ♦❢ r1 ❛♥❞ r2

r = λ1r1 + λ2r2.

❚❤❡ ♥✉♠❜❡rs λ1 ❛♥❞ λ2 ❛r❡ ❞❡✜♥❡❞ ✉♥✐q✉❡❧②✳✹✳ ❙❤♦✇ t❤❛t ✐❢ t❤r❡❡ ✈❡❝t♦rs r1✱ r2✱ r3 ❛r❡ ❧✐♥❡❛r❧② ✐♥❞❡♣❡♥❞❡♥t✱ t❤❡♥ ❛♥②

✈❡❝t♦r r ✐s ✉♥✐q✉❡❧② ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ t❤❡s❡ ✈❡❝t♦rs ✐♥ t❤❡ ❢♦r♠

r = λ1r1 + λ2r2 + λ3r3.

✶✼✳✺✳ ❙❈❆▲❆❘ P❘❖❉❯❈❚ ❖❋ ❱❊❈❚❖❘❙ ✶✾✾

❋✐❣✉r❡ ✶✼✳✼✿ ❙❝❛❧❛r ♣r♦❞✉❝t

✶✼✳✺ ❙❝❛❧❛r ♣r♦❞✉❝t ♦❢ ✈❡❝t♦rs

❚❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ t❤❡ ✈❡❝t♦rs a ❛♥❞ b ✐s ❞❡✜♥❡❞ ❛s t❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ t❤❡✈❡❝t♦rs ❡q✉❛❧ t♦ a ❛♥❞ b✱ r❡s♣❡❝t✐✈❡❧②✱ r❡❞✉❝❡❞ t♦ ❛ ❝♦♠♠♦♥ ♦r✐❣✐♥ ✭❋✐❣✳ ✶✼✳✼✮✳❚❤❡ s❝❛❧❛r ♣r♦❞✉❝t ♦❢ ❛ ✈❡❝t♦r a ❜② ❛ ✈❡❝t♦r b ✐s ❞❡✜♥❡❞ ❛s t❤❡ ♥✉♠❜❡r ab✇❤✐❝❤ ✐s ❡q✉❛❧ t♦ t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡ ♦❢ t❤❡ ✈❡❝t♦rs ❜② t❤❡❝♦s✐♥❡ ♦❢ t❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ t❤❡♠✳

❚❤❡ s❝❛❧❛r ♣r♦❞✉❝t ♣♦ss❡ss❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ♦❜✈✐♦✉s✱ ♣r♦♣❡rt✐❡s ✇❤✐❝❤ ❢♦❧❧♦✇❞✐r❡❝t❧② ❢r♦♠ ✐ts ❞❡✜♥✐t✐♦♥✿

✭✶✮ ab = ba❀

✭✷✮ a2 = aa = |a|2❀

✭✸✮ (λa)b = λ(ab)❀

✭✹✮ ✐❢ |e| = 1✱ t❤❡♥ (λe)(µe) = λµ❀

✭✺✮ t❤❡ s❝❛❧❛r ♣r♦❞✉❝t ♦❢ ✈❡❝t♦rs a ❛♥❞ b ✐s ❡q✉❛❧ t♦ ③❡r♦ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡✈❡❝t♦rs ❛r❡ ♠✉t✉❛❧❧② ♣❡r♣❡♥❞✐❝✉❧❛r ♦r ♦♥❡ ♦❢ t❤❡♠ ✐s ❡q✉❛❧ t♦ ③❡r♦✳

❚❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r a ♦♥ ❛ str❛✐❣❤t ❧✐♥❡ ✐s ❞❡✜♥❡❞ ❛s t❤❡ ✈❡❝t♦r a✇❤♦s❡ ✐♥✐t✐❛❧ ♣♦✐♥ts ✐s t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ t❤❡ ✐♥✐t✐❛❧ ♣♦✐♥t ♦❢ t❤❡ ✈❡❝t♦r a ❛♥❞✇❤♦s❡ t❡r♠✐♥❛❧ ♣♦✐♥t ✐s t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ t❤❡ t❡r♠✐♥❛❧ ♣♦✐♥t ♦❢ t❤❡ ✈❡❝t♦r a✳

❖❜✈✐♦✉s❧②✱ ❡q✉❛❧ ✈❡❝t♦rs ❤❛✈❡ ❡q✉❛❧ ♣r♦❥❡❝t✐♦♥s✱ t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ t❤❡ s✉♠♦❢ ✈❡❝t♦rs ✐s ❡q✉❛❧ t♦ t❤❡ s✉♠ ♦❢ t❤❡ ♣r♦❥❡❝t✐♦♥s ✭❋✐❣✳ ✶✼✳✽✮✳

❚❤❡ s❝❛❧❛r ♣r♦❞✉❝t ♦❢ ❛ ✈❡❝t♦r a ❜② ❛ ✈❡❝t♦r b ✐s ❡q✉❛❧ t♦ t❤❡ s❝❛❧❛r♣r♦❞✉❝t ♦❢ t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ t❤❡ ✈❡❝t♦r a ♦♥t♦ t❤❡ str❛✐❣❤t ❧✐♥❡ ❝♦♥t❛✐♥✐♥❣t❤❡ ✈❡❝t♦r b ❜② t❤❡ ✈❡❝t♦r b✳ ❚❤❡ ♣r♦♦❢ ✐s ♦❜✈✐♦✉s✳ ■t ✐s s✉✣❝✐❡♥t t♦ ♥♦t❡t❤❛t ab ❛♥❞ ab ❛r❡ ❡q✉❛❧ ❜② ❛❜s♦❧✉t❡ ✈❛❧✉❡ ❛♥❞ ❤❛✈❡ t❤❡ s❛♠❡ s✐❣♥✳

❚❤❡ s❝❛❧❛r ♣r♦❞✉❝t ♣♦ss❡ss❡s t❤❡ ❞✐str✐❜✉t✐✈❡ ♣r♦♣❡rt②✳ ◆❛♠❡❧② ❢♦r ❛♥②t❤r❡❡ ✈❡❝t♦rs a✱ b✱ c

(a+ b)c = ac+ bc.

✷✵✵ ❈❍❆P❚❊❘ ✶✼✳ ❱❊❈❚❖❘❙

❋✐❣✉r❡ ✶✼✳✽✿ Pr♦❥❡❝t✐♦♥ ♦❢ ✈❡❝t♦rs

❚❤❡ st❛t❡♠❡♥t ✐s ♦❜✈✐♦✉s ✐❢ ♦♥❡ ♦❢ t❤❡ ✈❡❝t♦rs ✐s ❡q✉❛❧ t♦ ③❡r♦✳ ▲❡t ❛❧❧ t❤❡✈❡❝t♦rs ❜❡ ♥♦♥✲③❡r♦✳ ❉❡♥♦t✐♥❣ ❜② a✱ b✱ a+ b t❤❡ ♣r♦❥❡❝t✐♦♥s ♦❢ t❤❡ ✈❡❝t♦rsa✱ b✱ ❛♥❞ a+ b ♦♥t♦ t❤❡ ❧✐♥❡ ❝♦♥t❛✐♥✐♥❣ t❤❡ ✈❡❝t♦r c✱ ✇❡ ❤❛✈❡

(a+ b)c = (a+ b)c = (a+ b)c,

ac+ bc = ac+ bc.

▲❡t e ❜❡ ❛ ✉♥✐t ✈❡❝t♦r ♣❛r❛❧❧❡❧ t♦ c✳ ❚❤❡♥ a✱ b✱ ❛♥❞ c ❛❧❧♦✇ t❤❡ r❡♣r❡s❡♥✲t❛t✐♦♥s a = λe✱ b = µe✱ c = νe✳ ❲❡ ♦❜t❛✐♥

(a+ b)c = (λe+ µe)νe = (λ+ µ)ν,

ac+ bc = λeνe+ µeνe = λν + µν.

❲❤❡♥❝❡

(a+ b)c = ac+ bc

❛♥❞✱ ❤❡♥❝❡

(a+ b)c = ac+ bc.

■♥ ❝♦♥❝❧✉s✐♦♥ ✇❡ ❛r❡ ❣♦✐♥❣ t♦ s❤♦✇ t❤❛t ✐❢ a✱ b✱ c ❛r❡ ♥♦♥✲③❡r♦ ✈❡❝t♦rs✇❤✐❝❤ ❛r❡ ♥♦t ♣❛r❛❧❧❡❧ t♦ ♦♥❡ ♣❧❛♥❡✱ t❤❡♥ ❢r♦♠ t❤❡ ❡q✉❛❧✐t✐❡s

ra = 0, rb = 0, rc = 0

✐❢ ❢♦❧❧♦✇s t❤❛t r = 0✳■♥❞❡❡❞✱ ✐❢ r 6= 0✱ t❤❡♥ ❢r♦♠ t❤❡ ❛❜♦✈❡ t❤r❡❡ ❡q✉❛❧✐t✐❡s ✐t ❢♦❧❧♦✇s t❤❛t t❤❡

✈❡❝t♦rs a✱ b✱ c ❛r❡ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ r✱ ❛♥❞ t❤❡r❡❢♦r❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ ♣❧❛♥❡♣❡r♣❡♥❞✐❝✉❧❛r t♦ r ✇❤✐❝❤ ✐s ✐♠♣♦ss✐❜❧❡✳

✶✼✳✻✳ ❊❳❊❘❈■❙❊❙ ✷✵✶

❋✐❣✉r❡ ✶✼✳✾✿ ❱❡❝t♦r ♣r♦❞✉❝t ♦❢ t✇♦ ✈❡❝t♦rs

✶✼✳✻ ❊①❡r❝✐s❡s

✶✳ ▲❡t A1, A2, . . . , An ❜❡ t❤❡ ✈❡rt✐❝❡s ♦❢ ❛ r❡❣✉❧❛r n✲❣♦♥✳ ❚❤❡♥−−−→A1A2 +−−−→

A2A3 + · · ·+−−−→AnA1 = 0✳ ❉r✐✈❡ ❢r♦♠ t❤✐s t❤❛t

1 + cos2π

n+ cos

4π

n+ · · ·+ cos

(2n− 2)π

n= 0,

sin2π

n+ sin

4π

n+ · · ·+ sin

(2n− 2)π

n= 0.

✷✳ ❙❤♦✇ t❤❛t ✐❢ a ❛♥❞ b ❛r❡ ♥♦♥✲③❡r♦ ❛♥❞ ♥♦♥✲♣❛r❛❧❧❡❧ ✈❡❝t♦rs✱ t❤❡♥λ2a2 + 2µλ(ab) + µ2b2 ≥ 0✱ t❤❡ ❡q✉❛❧✐t② t♦ ③❡r♦ t❛❦✐♥❣ ♣❧❛❝❡ ♦♥❧② ✐❢ λ = 0✱❛♥❞ µ = 0✳

✶✼✳✼ ❚❤❡ ✈❡❝t♦r ♣r♦❞✉❝t ♦❢ ✈❡❝t♦rs

❚❤❡ ✈❡❝t♦r ♣r♦❞✉❝t ♦❢ ❛ ✈❡❝t♦r a ❜② ❛ ✈❡❝t♦r b ✐s ❛ t❤✐r❞ ✈❡❝t♦r a×b ❞❡✜♥❡❞✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✇❛②✳ ■❢ ❛t ❧❡❛st ♦♥❡ ♦❢ t❤❡ ✈❡❝t♦rs a✱ b ✐s ❡q✉❛❧ t♦ ③❡r♦ ♦rt❤❡ ✈❡❝t♦rs ❛r❡ ♣❛r❛❧❧❡❧✱ t❤❡♥ a × b = 0✳ ✐♥ ♦t❤❡r ❝❛s❡s t❤✐s ✈❡❝t♦r ✭❜② ✐ts❛❜s♦❧✉t❡ ✈❛❧✉❡✮ ✐s ❡q✉❛❧ t♦ t❤❡ ❛r❡❛ ♦❢ t❤❡ ♣❛r❛❧❧❡❧♦❣r❛♠ ❝♦♥str✉❝t❡❞ ♦♥ t❤❡✈❡❝t♦rs a ❛♥❞ b ❛s s✐❞❡s ❛♥❞ ✐s ❞✐r❡❝t❡❞ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ♣❧❛♥❡ ❝♦♥t❛✐♥✐♥❣t❤✐s ♣❛r❛❧❧❡❧♦❣r❛♠ s♦ t❤❛t t❤❡ r♦t❛t✐♦♥ ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ❢r♦♠ a t♦ b ❛♥❞ t❤❡❞✐r❡❝t✐♦♥ ♦❢ a× b ❢♦r♠ ❛ ✏r✐❣❤t✲❤❛♥❞ s❝r❡✇✧ ✭❋✐❣✳ ✶✼✳✾✮✳

❋r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ✈❡❝t♦r ♣r♦❞✉❝t ✐t ❞✐r❡❝t❧② ❢♦❧❧♦✇s✿

✭✶✮ a× b = −b× a✱

✭✷✮ |a×b| = |a| |b| sin θ✱ ✇❤❡r❡ θ ✐s t❤❡ ❛♥❣❧❡ ❢♦r♠❡❞ ❜② t❤❡ ✈❡❝t♦rs a ❛♥❞ b❀

✷✵✷ ❈❍❆P❚❊❘ ✶✼✳ ❱❊❈❚❖❘❙

❋✐❣✉r❡ ✶✼✳✶✵✿ Pr♦❥❡❝t✐♦♥ ♦♥ ❛ ♣❧❛♥❡

❋✐❣✉r❡ ✶✼✳✶✶✿ Pr♦❥❡❝t✐♦♥ ♦♥ ♣❡r♣❡♥❞✐❝✉❧❛r ♣❧❛♥❡

✭✸✮ (λa)× b = λ(a× b)✳

❚❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r a ♦♥ ❛ ♣❧❛♥❡ ✐s ❞❡✜♥❡❞ ❛s t❤❡ ✈❡❝t♦r a′ ✇❤♦s❡✐♥✐t✐❛❧ ♣♦✐♥t ✐s t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ t❤❡ ✐♥✐t✐❛❧ ♣♦✐♥t ♦❢ t❤❡ ✈❡❝t♦r a ❛♥❞ ✇❤♦s❡t❡r♠✐♥❛❧ ♣♦✐♥t ✐s t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ t❤❡ t❡r♠✐♥❛❧ ♣♦✐♥t ♦❢ t❤❡ ✈❡❝t♦r a✳ ❖❜✈✐✲♦✉s❧②✱ ❡q✉❛❧ ✈❡❝t♦rs ❤❛✈❡ ❡q✉❛❧ ♣r♦❥❡❝t✐♦♥s ❛♥❞ t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ t❤❡ s✉♠ ♦❢✈❡❝t♦rs ✐s ❡q✉❛❧ t♦ t❤❡ s✉♠ ♦❢ t❤❡ ♣r♦❥❡❝t✐♦♥s ✭❋✐❣✳ ✶✼✳✶✵✮✳

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ t✇♦ ✈❡❝t♦rs a ❛♥❞ b✳ ▲❡t a′ ❞❡♥♦t❡ t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ t❤❡✈❡❝t♦r a ♦♥ t❤❡ ♣❧❛♥❡ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ✈❡❝t♦r b ✭❋✐❣✳ ✶✼✳✶✶✮✳ ❚❤❡♥

a× b = a′ × b.

❚❤❡ ♣r♦♦❢ ✐s ♦❜✈✐♦✉s✳ ■t ✐s s✉✣❝✐❡♥t t♦ ♠❡♥t✐♦♥ t❤❛t t❤❡ ✈❡❝t♦rs a × b

❛♥❞ a′ × b ❤❛✈❡ ❡q✉❛❧ ❛❜s♦❧✉t❡ ✈❛❧✉❡s ❛♥❞ ❛r❡ ✐♥ t❤❡ s❛♠❡ ❞✐r❡❝t✐♦♥✳❚❤❡ ✈❡❝t♦r ♣r♦❞✉❝t ♣♦ss❡ss❡s ❛ ❞✐str✐❜✉t✐✈❡ ♣r♦♣❡rt②✱ ✐✳❡✳ ❢♦r ❛♥② t❤❡r❡

✈❡❝t♦rs a✱ b✱ c(a+ b)× c = a× c+ b× c. (∗)

✶✼✳✼✳ ❚❍❊ ❱❊❈❚❖❘ P❘❖❉❯❈❚ ❖❋ ❱❊❈❚❖❘❙ ✷✵✸

❋✐❣✉r❡ ✶✼✳✶✷✿ ❉✐str✐❜✉t✐✈❡ ❧❛✇ ♦❢ ✈❡❝t♦r ♣r♦❞✉❝t

❚❤❡ ❛ss❡rt✐♦♥ ✐s ♦❜✈✐♦✉s ✐❢ c = 0✳ ■t ✐s t❤❡♥ ♦❜✈✐♦✉s t❤❛t t❤❡ ❡q✉❛❧✐t② (∗)✐s s✉✣❝✐❡♥t t♦ t❤❡ ❢♦r t❤❡ ❝❛s❡ |c| = 1✱ s✐♥❝❡ ✐♥ t❤❡ ❣❡♥❡r❛❧ ❝❛s❡ ✐t ✇✐❧❧ t❤❡♥❢♦❧❧♦✇ t❤❡ ❛❜♦✈❡ ♠❡♥t✐♦♥❡❞ ♣r♦♣❡rt② ✭✸✮✳

❙♦✱ ❧❡t |c| = 1✱ ❛♥❞ ❧❡t a′ ❛♥❞ b′ ❞❡♥♦t❡ t❤❡ ♣r♦❥❡❝t✐♦♥s ♦❢ t❤❡ ✈❡❝t♦rsa ❛♥❞ b ♦♥ t❤❡ ♣❧❛♥❡ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ✈❡❝t♦r c ✭❋✐❣✳ ✶✼✳✶✷✮✳ ❚❤❡♥ t❤❡✈❡❝t♦rs a′×c✱ b′×c ❛♥❞ (a′+b′)×c ❛r❡ ♦❜t❛✐♥❡❞ ❢r♦♠ t❤❡ ✈❡❝t♦rs a′✱ b′✱ ❛♥❞a′ + b′✱ r❡s♣❡❝t✐✈❡❧②✱ ❜② ❛ r♦t❛t✐♥❣ t❤r♦✉❣❤ ❛♥ ❛♥❣❧❡ ♦❢ 90◦✳ ❈♦♥s❡q✉❡♥t❧②✱

(a′ + b′)× c = a′ × c+ b′ × c.

❆♥❞ s✐♥❝❡

a′ × c = a× c, b′ × c = b× c,

(a′ + b′)× c = (a+ b)× c,

✇❡ ❣❡t(a+ b)× c = a× c+ b× c,

✇❤✐❝❤ ✇❛s r❡q✉✐r❡❞ t♦ ❜❡ ♣r♦✈❡❞✳▲❡t ✉s ♠❡♥t✐♦♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s✐♠♣❧❡ ✐❞❡♥t✐t② ✇❤✐❝❤ ✐s tr✉❡ ❢♦r ❛♥② ✈❡❝t♦rs

a ❛♥❞ b✿(a× b)2 = a2b2 − (ab)2.

■♥❞❡❡❞✱ ✐❢ θ ✐s t❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ t❤❡ ✈❡❝t♦rs a ❛♥❞ b✱ t❤❡♥ t❤✐s ✐♥❞❡♥t✐t②❡①♣r❡ss❡s t❤❛t

(|a| |b| sin θ)2 = |a|2|b|2 − (|a| |b| cos θ)2

❛♥❞✱ ❝♦♥s❡q✉❡♥t❧②✱ ✐s ♦❜✈✐♦✉s✳

✷✵✹ ❈❍❆P❚❊❘ ✶✼✳ ❱❊❈❚❖❘❙

✶✼✳✽ ❊①❡r❝✐s❡s

✶✳ ■❢ t❤❡ ✈❡❝t♦rs a ❛♥❞ b ❛r❡ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ✈❡❝t♦r c✱ t❤❡♥

(a× b)× c = 0.

❙❤♦✇ t❤✐s✳✷✳ ■❢ t❤❡ ✈❡❝t♦r b ✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ c✱ ❛♥❞ t❤❡ ✈❡❝t♦r a ✐s ♣❛r❛❧❧❡❧ t♦

t❤❡ ✈❡❝t♦r c✱ t❤❡♥(a× b)× c = b(ac).

❙❤♦✇ t❤✐s✳✸✳ ❋♦r ❛♥ ❛r❜✐tr❛r② ✈❡❝t♦r a ❛♥❞ ❛ ✈❡❝t♦r b ♣❡r♣❡♥❞✐❝✉❧❛r t♦ c

(a× b)× c = b(ac).

❙❤♦✇ t❤✐s✳✹✳ ❙❤♦✇ t❤❛t ❢♦r ❛♥② t❤r❡❡ ✈❡❝t♦rs a✱ b✱ c

(a× b)× c = b(ac)− a(bc).

✺✳ ❋✐♥❞ t❤❡ ❛r❡❛ ♦❢ t❤❡ ❜❛s❡ ♦❢ ❛ tr✐❛♥❣✉❧❛r ♣②r❛♠✐❞ ✇❤♦s❡ ❧❛t❡r❛❧ ❡❞❣❡s❛r❡ ❡q✉❛❧ t♦ l✱ t❤❡ ✈❡rt❡① ❛♥❣❧❡s ❜❡✐♥❣ ❡q✉❛❧ t♦ α✱ β✱ γ✳

✶✼✳✾ ❚❤❡ tr✐♣❧❡ ♣r♦❞✉❝t ♦❢ ✈❡❝t♦rs

❚❤❡ tr✐♣❧❡ ✭s❝❛❧❛r✮ ♣r♦❞✉❝t ♦❢ ✈❡❝t♦rs a✱ b✱ c ✐s t❤❡ ♥✉♠❜❡r

(abc) = (a× b)c. (∗)

❖❜✈✐♦✉s❧②✱ t❤❡ tr✐♣❧❡ ♣r♦❞✉❝t ✐s ❡q✉❛❧ t♦ ③❡r♦ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ♦♥❡ ♦❢ t❤❡✈❡❝t♦rs ✐s ❡q✉❛❧ t♦ ③❡r♦ ♦r ❛❧❧ t❤r❡❡ ✈❡❝t♦rs ❛r❡ ♣❛r❛❧❧❡❧ t♦ ♦♥❡ ♣❧❛♥❡✳

❚❤❡ ♥✉♠❡r✐❝❛❧ ✈❛❧✉❡ ♦❢ t❤❡ tr✐♣❧❡ ♣r♦❞✉❝t ♦❢ ♥♦♥✲③❡r♦ ✈❡❝t♦rs a✱ b✱ c ✇❤✐❝❤❛r❡ ♥♦t ♣❛r❛❧❧❡❧ t♦ ♦♥❡ ♣❧❛♥❡ ✐s ❡q✉❛❧ t♦ t❤❡ ✈♦❧✉♠❡ ♦❢ t❤❡ ♣❛r❛❧❧❡❧❡♣✐♣❡❞ ♦❢✇❤✐❝❤ t❤❡ ✈❡❝t♦rs a✱ b✱ c ❛r❡ ❝♦t❡r♠✐♥❛❧ s✐❞❡s ✭❋✐❣✳ ✶✼✳✶✸✮✳

■♥❞❡❡❞✱ a× b = Se✱ ✇❤❡r❡ S ✐s t❤❡ ❛r❡ ♦❢ t❤❡ ❜❛s❡ ♦❢ t❤❡ ♣❛r❛❧❧❡❧❡♣✐♣❡❞❝♦♥str✉❝t❡❞ ♦♥ t❤❡ ✈❡❝t♦rs a✱ b✱ ❛♥❞ e ✐s t❤❡ ✉♥✐t ✈❡❝t♦r ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡❜❛s❡✳ ❋✉rt❤❡r✱ ec ✐s ❡q✉❛❧ ✉♣ t♦ ❛ s✐♥❣❧❡ t♦ t❤❡ ❛❧t✐t✉❞❡ ♦❢ t❤❡ ♣❛r❛❧❧❡❧❡♣✐♣❡❞❞r♦♣♣❡❞ ♦♥t♦ t❤❡ ♠❡♥t✐♦♥❡❞ ❜❛s❡✳ ❈♦♥s❡q✉❡♥t❧②✱ ✉♣ t♦ ❛ s✐❣♥✱ (abc) ✐s ❡q✉❛❧t♦ t❤❡ ✈♦❧✉♠❡ ♦❢ t❤❡ ♣❛r❛❧❧❡❧❡♣✐♣❡❞ ❝♦♥str✉❝t❡❞ ♦♥ t❤❡ ✈❡❝t♦rs a✱ b✱ ❛♥❞ c✳❚❤❡ tr✐♣❧❡ ♣r♦❞✉❝t ♣♦ss❡ss❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt②

(abc) = a(b× c). (∗∗)

✶✼✳✶✵✳ ❊❳❊❘❈■❙❊❙ ✷✵✺

❋✐❣✉r❡ ✶✼✳✶✸✿ ▼❡❛♥✐♥❣ ♦❢ tr✐♣❧❡ ♣r♦❞✉❝t

■t ✐s s✉✣❝✐❡♥t t♦ ♥♦t❡ t❤❛t t❤❡ r✐❣❤t✲❤❛♥❞ ❛♥❞ t❤❡ ❧❡❢t✲❤❛♥❞ ♠❡♠❜❡rs ❛r❡❡q✉❛❧ ❜② ❛❜s♦❧✉t❡ ✈❛❧✉❡ ❛♥❞ ❤❛✈❡ t❤❡ s❛♠❡ s✐❣♥✳ ❋r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥ (∗) ♦❢t❤❡ tr✐♣❧❡ ♣r♦❞✉❝t ❛♥❞ t❤❡ ♣r♦♣❡rt② (∗∗) ✐t ❢♦❧❧♦✇s t❤❛t ❛♥ ✐♥t❡r❝❤❛♥❣❡ ♦❢ ❛♥②t✇♦ ❢❛❝t♦rs r❡✈❡rs❡s t❤❡ s✐❣♥ ♦❢ t❤❡ tr✐♣❧❡ ♣r♦❞✉❝t✳ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡ tr✐♣❧❡♣r♦❞✉❝t ✐s ❡q✉❛❧ t♦ ③❡r♦ ✐❢ t✇♦ ❢❛❝t♦rs ❛r❡ ❡q✉❛❧ t♦ ❡❛❝❤ ♦t❤❡r✳

✶✼✳✶✵ ❊①❡r❝✐s❡s

✶✳ ◆♦t✐♥❣ t❤❛t

((a× b)× c)d = (a× b)(c× d),

❞❡r✐✈❡ t❤❡ ✐❞❡♥t✐t②

(a× b)(c× d) =

∣

∣

∣

∣

ac ad

bc bd

∣

∣

∣

∣

.

✷✳ ❲✐t❤ t❤❡ ❛✐❞ ♦❢ t❤❡ ✐❞❡♥t✐t②

(a× b)(c× b) = (ac)b2 − (ab)(bc)

❞❡r✐✈❡ t❤❡ ❢♦r♠✉❧❛ ♦❢ s♣❤❡r✐❝❛❧ tr✐❣♦♥♦♠❡tr② ✇❤❡r❡ α✱ β✱ γ ❛r❡ t❤❡ s✐❞❡s ♦❢❛ tr✐❛♥❣❧❡ ♦♥ t❤❡ ✉♥✐t s♣❤❡r❡✱ ❛♥❞ B ✐s t❤❡ ❛♥❣❧❡ ♦❢ t❤✐s tr✐❛♥❣❧❡ ♦♣♣♦s✐t❡ t♦t❤❡ s✐❞❡ β✳

✸✳ ❉❡r✐✈❡ t❤❡ ✐❞❡♥t✐t②

(a× b)(c× d) = b(acd)− a(bcd).

✷✵✻ ❈❍❆P❚❊❘ ✶✼✳ ❱❊❈❚❖❘❙

❈❤❛♣t❡r ✶✽

❘❡❝t❛♥❣✉❧❛r ❈❛rt❡s✐❛♥

❈♦♦r❞✐♥❛t❡s ✐♥ ❙♣❛❝❡

✶✽✳✶ ❈❛rt❡s✐❛♥ ❝♦♦r❞✐♥❛t❡s

▲❡t ✉s ❞r❛✇ ❢r♦♠ ❛♥ ❛r❜✐tr❛r② ♣♦✐♥t O ✐♥ s♣❛❝❡ t❤r❡❡ str❛✐❣❤t ❧✐♥❡s Ox✱ Oy✱Oz ♥♦t ❧②✐♥❣ ✐♥ ♦♥❡ ♣❧❛♥❡✱ ❛♥❞ ❧❛② ♦✛ ♦♥ ❡❛❝❤ ♦❢ t❤❡♠ ❢r♦♠ t❤❡ ♣♦✐♥t O t❤r❡❡♥♦♥✲③❡r♦ ✈❡❝t♦rs ex✱ ey✱ ez ✭❋✐❣✳ ✶✽✳✶✮✳ ❆❝❝♦r❞✐♥❣ t♦ ❙❡❝t✐♦♥ ✶✼✳✻✱ ❛♥② ✈❡❝t♦r−→OA ❛❧❧♦✇s ❛ ✉♥✐q✉❡ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❢♦r♠

−→OA = xex + yey + zez.

❚❤❡ ♥✉♠❜❡rs x✱ y z ❛r❡ ❝❛❧❧❡❞ t❤❡ ❈❛rt❡s✐❛♥ ❝♦♦r❞✐♥❛t❡s ♦❢ ❛ ♣♦✐♥t A✳❚❤❡ str❛✐❣❤t ❧✐♥❡s Ox✱ Oy✱ Oz ❛r❡ t❡r♠❡❞ t❤❡ ❝♦♦r❞✐♥❛t❡ ❛①❡s✿ Ox ✐s t❤❡

x✲❛①✐s✱ Oy ✐s t❤❡ y✲❛①✐s✱ ❛♥❞ Oz ✐s t❤❡ z✲❛①✐s✳ ❚❤❡ ♣❧❛♥❡s Oxy✱ Oyz✱ Oxz❛r❡ ❝❛❧❧❡❞ t❤❡ ❝♦♦r❞✐♥❛t❡ ♣❧❛♥❡s✿ Oxy ✐s t❤❡ xy✲♣❧❛♥❡✱ Oyz ✐s t❤❡ yz✲♣❧❛♥❡✱❛♥❞ Oxz ✐s t❤❡ xz✲♣❧❛♥❡✳

❋✐❣✉r❡ ✶✽✳✶✿ ❈♦♦r❞✐♥❛t❡ ❛①❡s ✐♥ s♣❛❝❡

✷✵✼

✷✵✽❈❍❆P❚❊❘ ✶✽✳ ❘❊❈❚❆◆●❯▲❆❘ ❈❆❘❚❊❙■❆◆ ❈❖❖❘❉■◆❆❚❊❙ ■◆ ❙P❆❈❊

❋✐❣✉r❡ ✶✽✳✷✿ ❈♦♦r❞✐♥❛t❡s ✐♥ s♣❛❝❡

❊❛❝❤ ♦❢ t❤❡ ❝♦♦r❞✐♥❛t❡ ❛①❡s ✐s ❞✐✈✐❞❡❞ ❜② t❤❡ ♣♦✐♥t O ✭✐✳❡✳ ❜② t❤❡ ♦r✐❣✐♥♦❢ ❝♦♦r❞✐♥❛t❡s✮ ✐♥t♦ t✇♦ s❡♠✐✲❛①❡s✳ ❚❤♦s❡ ♦❢ t❤❡ s❡♠✐✲❛①❡s ✇❤♦s❡ ❞✐r❡❝t✐♦♥s❝♦✐♥❝✐❞❡ ✇✐t❤ t❤❡ ❞✐r❡❝t✐♦♥s ♦❢ t❤❡ ✈❡❝t♦rs ex✱ ey✱ ez ❛r❡ s❛✐❞ t♦ ❜❡ ♣♦s✐t✐✈❡✱t❤❡ ♦t❤❡rs ❜❡✐♥❣ ♥❡❣❛t✐✈❡✳ ❚❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ t❤✉s ♦❜t❛✐♥❡❞ ✐s ❝❛❧❧❡❞r✐❣❤t✲❤❛♥❞❡❞ ✐❢ (exeyez) > 0✱ ❛♥❞ ❧❡❢t✲❤❛♥❞❡❞ ✐❢ (exeyez) < 0✳

●❡♦♠❡tr✐❝❛❧❧② t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ♣♦✐♥t A ❛r❡ ♦❜t❛✐♥❡❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣✇❛②✳ ❲❡ ❞r❛✇ t❤r♦✉❣❤ t❤❡ ♣♦✐♥t A ❛ ♣❧❛♥❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ yz✲♣❧❛♥❡✳ ■t✐♥t❡rs❡❝ts t❤❡ x✲❛①✐s ❛t ❛ ♣♦✐♥t Ax ✭❋✐❣✳ ✶✽✳✷✮✳ ❚❤❡♥ t❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡ ♦❢t❤❡ ❝♦♦r❞✐♥❛t❡ x ♦❢ t❤❡ ♣♦✐♥t A ✐s ❡q✉❛❧ t♦ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❧✐♥❡ s❡❣♠❡♥tOAx ❛s ♠❡❛s✉r❡❞ ❜② t❤❡ ✉♥✐t ❧❡♥❣t❤ |ex|✳

■t ✐s ♣♦s✐t✐✈❡ ✐❢ Ax ❜❡❧♦♥❣s t♦ t❤❡ ♣♦s✐t✐✈❡ s❡♠✐✲❛①✐s x✱ ❛♥❞ ✐s ♥❡❣❛t✐✈❡ ✐❢Ax ❜❡❧♦♥❣s t♦ t❤❡ ♥❡❣❛t✐✈❡ s❡♠✐✲❛①✐s x✳ ❚♦ ♠❛❦❡ s✉r❡ ♦❢ t❤✐s ✐s s✉✣❝✐❡♥t t♦r❡❝❛❧❧ ❤♦✇ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ✈❡❝t♦r

−→OA r❡❧❛t✐✈❡ t♦ t❤❡ ❜❛s✐s ex✱ ey✱ ez ❛r❡

❞❡t❡r♠✐♥❡❞✳ ❚❤❡ ♦t❤❡r t✇♦ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ♣♦✐♥t ✭y ❛♥❞ z✮ ❛r❡ ❞❡t❡r♠✐♥❡❞❜② ❛ s✐♠✐❧❛r ❝♦♥str✉❝t✐♦♥✳

■❢ t❤❡ ❝♦♦r❞✐♥❛t❡ ❛①❡s ❛r❡ ♠✉t✉❛❧❧② ♣❡r♣❡♥❞✐❝✉❧❛r✱ ❛♥❞ ex✱ ey✱ ez ❛r❡t❤❡ ✉♥✐t ✈❡❝t♦rs✱ t❤❡♥ t❤❡ ❝♦♦r❞✐♥❛t❡s ❛r❡ ❝❛❧❧❡❞ t❤❡ r❡❝t❛♥❣✉❧❛r ❈❛rt❡s✐❛♥❝♦♦r❞✐♥❛t❡s✳

❈❛rt❡s✐❛♥ ❝♦♦r❞✐♥❛t❡s ♦♥ t❤❡ ♣❧❛♥❡ ❛r❡ ✐♥tr♦❞✉❝❡❞ ✐♥ ❛ s✐♠✐❧❛r ✇❛②✳ ◆❛♠❡❧②✱✇❡ ❞r❛✇ ❢r♦♠ t❤❡ ♣♦✐♥t O ✭✐✳❡✳ ❢r♦♠ t❤❡ ♦r✐❣✐♥ ♦❢ ❝♦♦r❞✐♥❛t❡s✮ t✇♦ ❛r❜✐tr❛r②str❛✐❣❤t ❧✐♥❡s Ox ❛♥❞ Oy ✭t❤❡ ❝♦♦r❞✐♥❛t❡ ❛①❡s✮ ❛♥❞ ❧❛② ♦✛ ♦♥ ❡❛❝❤ ❛①✐s ✭❢r♦♠t❤❡ ♣♦✐♥t O✮ ❛ ♥♦♥✲③❡r♦ ✈❡❝t♦r✳ ❚❤✉s ✇❡ ♦❜t❛✐♥ t❤❡ ✈❡❝t♦rs ex ❛♥❞ ey✳ ❚❤❡❈❛rt❡s✐❛♥ ❝♦♦r❞✐♥❛t❡s ♦❢ ❛♥ ❛r❜✐tr❛r② ♣♦✐♥t A ❛r❡ t❤❡♥ ❞❡t❡r♠✐♥❡❞ ❛s t❤❡❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ✈❡❝t♦r

−→OA r❡❧❛t✐✈❡ t♦ t❤❡ ❜❛s✐s ex✱ ey✳

❖❜✈✐♦✉s❧②✱ ✐❢ t❤❡ ❝♦♦r❞✐♥❛t❡ ❛①❡s ❛r❡ ♠✉t✉❛❧❧② ♣❡r♣❡♥❞✐❝✉❧❛r✱ ❛♥❞ ex✱ey ❛r❡ ✉♥✐t ✈❡❝t♦rs✱ t❤❡♥ t❤❡ ❝♦♦r❞✐♥❛t❡s ❞❡✜♥❡❞ ✐♥ t❤✐s ✇❛② ❝♦✐♥❝✐❞❡ ✇✐t❤t❤♦s❡ ✐♥tr♦❞✉❝❡❞ ✐♥ ❙❡❝t✐♦♥ ✶✺✳✶ ❛♥❞ ❛r❡ ❝❛❧❧❡❞ t❤❡ r❡❝t❛♥❣✉❧❛r ❈❛rt❡s✐❛♥

✶✽✳✷✳ ❊❳❊❘❈■❙❊❙ ✷✵✾

❝♦♦r❞✐♥❛t❡s✳❇❡❧♦✇✱ ❛s ❛ r✉❧❡✱ ✇❡ s❤❛❧❧ ✉s❡ t❤❡ r❡❝t❛♥❣✉❧❛r ❈❛rt❡s✐❛♥ ❝♦♦r❞✐♥❛t❡s✳ ■❢

♦t❤❡r✇✐s❡✱ ❡❛❝❤ ❝❛s❡ ✇✐❧❧ ❜❡ s✉♣♣❧✐❡❞ ✇✐t❤ ❛ s♣❡❝✐❛❧ ♠❡♥t✐♦♥✳

✶✽✳✷ ❊①❡r❝✐s❡s

✶✳ ❲❤❡r❡ ❛r❡ t❤❡ ♣♦✐♥ts ✐♥ s♣❛❝❡ ❧♦❝❛t❡❞ ✐❢✿ ✭❛✮ x = 0❀ ✭❜✮ y = 0❀ ✭❝✮z = 0❀ ✭❞✮ x = 0, y = 0❀ ✭❡✮ y = 0, z = 0❀ ✭❢✮ z = 0, x = 0❄

✷✳ ❍♦✇ ♠❛♥② ♣♦✐♥ts ✐♥ s♣❛❝❡ s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s

|x| = a, |y| = b, |z| = c, ✐❢ abc 6= 0?

✸✳ ❲❤❡r❡ ❛r❡ t❤❡ ♣♦✐♥ts ✐♥ s♣❛❝❡ s✐t✉❛t❡❞ ✐❢

|x| < a, |y| < b, |z| < c?

✹✳ ▲❡t A ❜❡ ❛ ✈❡rt❡① ♦❢ ❛ ♣❛r❛❧❧❡❧❡♣✐♣❡❞✱ A1✱ A2✱ A3 t❤❡ ✈❡rt✐❝❡s ❛❞❥❛❝❡♥tt♦ A✱ ✐✳❡✳ t❤❡ ❡♥❞✲♣♦✐♥ts ♦❢ t❤❡ ❡❞❣❡s ❡♠❛♥❛t✐♥❣ ❢r♦♠ A✳ ❋✐♥❞ t❤❡ ❝♦♦r❞✐♥❛t❡s♦❢ ❛❧❧ t❤❡ ✈❡rt✐❝❡s ♦❢ t❤❡ ♣❛r❛❧❧❡❧❡♣✐♣❡❞✱ t❛❦✐♥❣ t❤❡ ✈❡rt❡① A ❢♦r t❤❡ ♦r✐❣✐♥ ❛♥❞t❤❡ ✈❡rt✐❝❡s A1✱ A2✱ A3 ❢♦r t❤❡ ❡♥❞✲♣♦✐♥ts ♦❢ t❤❡ ❜❛s✐s ✈❡❝t♦rs✳

✺✳ ❋✐♥❞ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ♣♦✐♥t ✐♥t♦ ✇❤✐❝❤ t❤❡ ♣♦✐♥t (x, y, z) ❣♦❡s✇❤❡♥ r♦t❛t❡❞ ❛❜♦✉t t❤❡ str❛✐❣❤t ❧✐♥❡ ❥♦✐♥✐♥❣ t❤❡ ♣♦✐♥t A0(a, b, c) t♦ t❤❡ ♦r✐❣✐♥t❤r♦✉❣❤ ❛♥ ❛♥❣❧❡ ♦❢ α = π/2✳ ❚❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ✐s r❡❝t❛♥❣✉❧❛r✳

✻✳ ❙♦❧✈❡ ❊①❡r❝✐s❡s ✺ ❢♦r ❛♥ ❛r❜✐tr❛r② α✳

✶✽✳✸ ❊❧❡♠❡♥t❛r② ♣r♦❜❧❡♠s ♦❢ s♦❧✐❞ ❛♥❛❧②t✐❝ ❣❡✲

♦♠❡tr②

▲❡t t❤❡r❡ ❜❡ ✐♥tr♦❞✉❝❡❞ ✐♥ s♣❛❝❡ ❈❛rt❡s✐❛♥ ❝♦♦r❞✐♥❛t❡s xyz ❛♥❞ ❧❡tA1(x1, y1, z1)❛♥❞ A2(x2, y2, z2) ❜❡ t✇♦ ❛r❜✐tr❛r② ♣♦✐♥ts ✐s ♣❛❝❡✳ ❋✐♥❞ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡♣♦✐♥t A ✇❤✐❝❤ ❞✐✈✐❞❡s t❤❡ ❧✐♥❡ s❡❣♠❡♥t A1A2 ✐♥ t❤❡ r❛t✐♦ λ1 : λ2 ✭❋✐❣✳ ✶✽✳✸✮✳

❚❤❡ ✈❡❝t♦rs−−→A1A ❛♥❞

−−→AA2 ❛r❡ ✐♥ t❤❡ s❛♠❡ ❞✐r❡❝t✐♦♥✱ ❛♥❞ t❤❡✐r ❛❜s♦❧✉t❡

✈❛❧✉❡s ❛r❡ ❛s λ1 : λ2✳ ❈♦♥s❡q✉❡♥t❧②✱

λ2

−−→A1A− λ1

−−→AA2 = 0,

♦rλ2(

−→OA−−−→

OA1)− λ2(−−→OA2 −

−→OA) = 0.

✷✶✵❈❍❆P❚❊❘ ✶✽✳ ❘❊❈❚❆◆●❯▲❆❘ ❈❆❘❚❊❙■❆◆ ❈❖❖❘❉■◆❆❚❊❙ ■◆ ❙P❆❈❊

❋✐❣✉r❡ ✶✽✳✸✿ ❉✐✈✐s✐♦♥ ♦❢ ❛ s❡❣♠❡♥t ✐♥ s♣❛❝❡

❲❤❡♥❝❡−→OA =

λ2

−−→OA1 + λ1

−−→OA2

λ1 + λ2

.

❙✐♥❝❡ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ♣♦✐♥ts A(x, y, z) ❛r❡ t❤❡ s❛♠❡ ❛s t❤❡ ❝♦♦r❞✐✲♥❛t❡s ♦❢ t❤❡ ✈❡❝t♦r

−→OA✱ ✇❡ ❤❛✈❡

x =λ2x1 + λ1x2

λ1 + λ2

,

y =λ2y1 + λ1y2λ1 + λ2

,

z =λ2z1 + λ1z2λ1 + λ2

.

▲❡t t❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ❜❡ r❡❝t❛♥❣✉❧❛r✳ ❊①♣r❡ss t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥t❤❡ ♣♦✐♥ts A1 ❛♥❞ A2 ✐♥ t❡r♠s ♦❢ t❤❡✐r ❝♦♦r❞✐♥❛t❡s✳

❚❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ♣♦✐♥ts A1 ❛♥❞ A2 ✐s ❡q✉❛❧ t♦ t❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡♦❢ t❤❡ ✈❡❝t♦r

−−−→A1A2 ✭❋✐❣✳ ✶✽✳✹✮✳ ❲❡ ❤❛✈❡

−−−→A1A2 =

−−→OA2 −

−−→OA1 = ex(x2 − x1) + ey(y2 − y1) + ez(z2 − z1).

❲❤❡♥❝❡(A1A2)

2 = (x2 − x1)2 + (y2 − y1)

2 + (z2 − z1)2.

❊①♣r❡ss t❤❡ ❛r❡❛ ♦❢ ❛ tr✐❛♥❣❧❡ ✐♥ t❤❡ xy✲♣❧❛♥❡ ✐♥ t❡r♠s ♦❢ t❤❡ ❝♦♦r❞✐♥❛t❡s♦❢ ✐ts ✈❡rt✐❝❡s✿ A1(x1, y1, 0)✱ A2(x2, y2, 0)✱ ❛♥❞ A3(x3, y3, 0)✳

❚❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡ ♦❢ t❤❡ ✈❡❝t♦r−−−→A1A2 ×

−−−→A1A3 ✐s ❡q✉❛❧ t♦ t✇✐❝❡ t❤❡ ❛r❡❛

♦❢ t❤❡ tr✐❛♥❣❧❡ A1A2A3❀

−−−→A1A2 ×

−−−→A1A3 = ez

∣

∣

∣

∣

x2 − x1 y2 − y1x3 − x1 y3 − y1

∣

∣

∣

∣

.

✶✽✳✹✳ ❊❳❊❘❈■❙❊❙ ✷✶✶

❋✐❣✉r❡ ✶✽✳✹✿ ❚❤❡ ❞✐st❛♥❝❡ ♦❢ t✇♦ ♣♦✐♥ts

❈♦♥s❡q✉❡♥t❧②✱ t❤❡ ❛r❡❛ ♦❢ t❤❡ tr✐❛♥❣❧❡

S =1

2

∣

∣

∣

∣

x2 − x1 y2 − y1x3 − x1 y3 − y1

∣

∣

∣

∣

.

❊①♣r❡ss t❤❡ ✈♦❧✉♠❡ ♦❢ ❛ t❡tr❛❤❡❞r♦♥ A1A2A3A4 ✐♥ t❡r♠s ♦❢ t❤❡ ❝♦♦r❞✐♥❛t❡s♦❢ ✐ts ✈❡rt✐❝❡s✳

❚❤❡ tr✐♣❧❡ s❝❛❧❛r ♣r♦❞✉❝t ♦❢ t❤❡ ✈❡❝t♦rs−−−→A1A2✱

−−−→A1A3✱

−−−→A1A4 ✐s ❡q✉❛❧ ✭✉♣ t♦

❛ s✐❣♥✮ t♦ t❤❡ ✈♦❧✉♠❡ ♦❢ t❤❡ ♣❛r❛❧❧❡❧❡♣✐♣❡❞ ❝♦♥str✉❝t❡❞ ♦♥ t❤❡s❡ ✈❡❝t♦rs ❛♥❞✱❝♦♥s❡q✉❡♥t❧②✱ t♦ s✐① t✐♠❡s t❤❡ ✈♦❧✉♠❡ ♦❢ t❤❡ t❡tr❛❤❡❞r♦♥ A1A2A3A4✳ ❍❡♥❝❡

V =1

6

∣

∣

∣

∣

∣

∣

x2 − x1 y2 − y1 z2 − z1x3 − x1 y3 − y1 z3 − z1x4 − x1 y4 − y1 z4 − z1

∣

∣

∣

∣

∣

∣

.

✶✽✳✹ ❊①❡r❝✐s❡s

✶✳ ❋✐♥❞ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ ❈❛rt❡s✐❛♥❝♦♦r❞✐♥❛t❡s ✐❢ t❤❡ ♣♦s✐t✐✈❡ s❡♠✐✲❛①❡s ❢♦r♠ ♣❛✐r✇✐s❡ t❤❡ ❛♥❣❧❡s α✱ β✱ γ✱ ❛♥❞ex✱ ey✱ ez ❛r❡ ✉♥✐t ✈❡❝t♦rs✳

✷✳ ❋✐♥❞ t❤❡ ❝❡♥tr❡ ♦❢ ❛ s♣❤❡r❡ ❝✐r❝✉♠sr✐❜❡❞ ❛❜♦✉t ❛ t❡tr❛❤❡❞r♦♥ ✇✐t❤ t❤❡✈❡rt✐❝❡s (a, 0, 0)✱ (0, b, 0)✱ (0, 0, c)✱ (0, 0, 0)✳

✸✳ Pr♦✈❡ t❤❛t t❤❡ str❛✐❣❤t ❧✐♥❡s ❥♦✐♥✐♥❣ t❤❡ ♠✐❞✲♣♦✐♥ts ♦❢ t❤❡ ♦♣♣♦s✐t❡❡❞❣❡s ♦❢ ❛ t❡tr❛❤❡❞r♦♥ ✐♥t❡rs❡❝t ❛t ♦♥❡ ♣♦✐♥t✳ ❊①♣r❡ss t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤✐s♣♦♥✐♥t ✐♥ t❡r♠s ♦❢ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ✈❡rt✐❝❡s ♦❢ t❤❡ t❡tr❛❤❡❞r♦♥✳

✹✳ Pr♦✈❡ t❤❛t t❤❡ str❛✐❣❤t ❧✐♥❡s ❥♦✐♥✐♥❣ t❤❡ ✈❡rt✐❝❡s ♦❢ ❛ t❡tr❛❤❡❞r♦♥ t♦t❤❡ ❝❡♥tr♦✐❞s ♦❢ t❤❡ ♦♣♣♦s✐t❡ ❢❛❝❡s ✐♥t❡rs❡❝t ❛t ♣♦✐♥t✳ ❊①♣r❡ss ✐ts ❝♦♦r❞✐♥❛t❡s✐♥ t❡r♠s ♦❢ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ✈❡rt✐❝❡s ♦❢ t❤❡ t❡tr❛❤❡❞r♦♥✳

✷✶✷❈❍❆P❚❊❘ ✶✽✳ ❘❊❈❚❆◆●❯▲❆❘ ❈❆❘❚❊❙■❆◆ ❈❖❖❘❉■◆❆❚❊❙ ■◆ ❙P❆❈❊

✶✽✳✺ ❊q✉❛t✐♦♥s ♦❢ ❛ s✉r❢❛❝❡ ❛♥❞ ❛ ❝✉r✈❡ ✐♥ s♣❛❝❡

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ s✉r❢❛❝❡✳❚❤❡ ❡q✉❛t✐♦♥

f(x, y, z) = 0 (∗)✐s ❝❛❧❧❡❞ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ s✉r❢❛❝❡ ✐♥ ✐♠♣❧✐❝✐t ❢♦r♠ ✐❢ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ❛♥②♣♦✐♥t ♦❢ t❤❡ s✉r❢❛❝❡ s❛t✐s❢② t❤✐s ❡q✉❛t✐♦♥✳ ❆♥❞ ❝♦♥✈❡rs❡❧②✱ ❛♥② t❤r❡❡ ♥✉♠❜❡rsx✱ y✱ z s❛t✐s❢②✐♥❣ t❤❡ ❡q✉❛t✐♦♥ (∗) r❡♣r❡s❡♥t t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ♦♥❡ ♦❢ t❤❡♣♦✐♥ts ♦❢ t❤❡ s✉r❢❛❝❡✳

❚❤❡ s②st❡♠ ♦❢ ❡q✉❛t✐♦♥s

x = f1(u, v), y = f2(u, v), z = f3(u, v), (∗∗)

s♣❡❝✐❢②✐♥❣ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ♣♦✐♥ts ♦❢ t❤❡ s✉r❢❛❝❡s ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦♣❛r❛♠❡t❡rs (u, v) ✐s ❝❛❧❧❡❞ t❤❡ ♣❛r❛♠❡tr✐❝ ❡q✉❛t✐♦♥ ♦❢ ❛ s✉r❢❛❝❡✳

❊❧✐♠✐♥❛t✐♥❣ t❤❡ ♣❛r❛♠❡t❡rs u✱ v ❢r♦♠ t❤❡ s②st❡♠ (∗∗)✱ ✇❡ ❝❛♥ ♦❜t❛✐♥ t❤❡✐♠♣❧✐❝✐t ❡q✉❛t✐♦♥ ♦❢ ❛ s✉r❢❛❝❡✳

❋♦r♠ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛♥ ❛r❜✐tr❛r② s♣❤❡r❡ ✐♥ t❤❡ r❡❝t❛♥❣✉❧❛r ❈❛rt❡s✐❛♥❝♦♦r❞✐♥❛t❡s xyz✳

▲❡t (x0, y0, z0) ❜❡ t❤❡ ❝❡♥tr❡ ♦❢ t❤❡ s♣❤❡r❡✱ ❛♥❞ R ✐ts r❛❞✐✉s✳ ❊❛❝❤ ♣♦✐♥t(x, y, z) ♦❢ t❤❡ s♣❤❡r❡ ✐s ❧♦❝❛t❡❞ ❛t ❛ ❞✐st❛♥❝❡ R ❢r♦♠ t❤❡ ❝❡♥tr❡✱ ❛♥❞✱ ❝♦♥s❡✲q✉❡♥t❧②✱ s❛t✐s✜❡s t❤❡ ❡q✉❛t✐♦♥

(x− x0)2 + (y − y0)

2 + (z − z0)2 −R2 = 0. (∗ ∗ ∗)

❈♦♥✈❡rs❡❧②✱ ❛♥② ♣♦✐♥t (x, y, z) s❛t✐s❢②✐♥❣ t❤❡ ❡q✉❛t✐♦♥ (∗∗∗) ✐s ❢♦✉♥❞ ❛t ❛ ❞✐s✲t❛♥❝❡ R ❢r♦♠ (x0, y0, z0) ❛♥❞✱ ❝♦♥s❡q✉❡♥t❧②✱ ❜❡❧♦♥❣ t♦ t❤❡ s♣❤❡r❡✳ ❆❝❝♦r❞✐♥❣t♦ t❤❡ ❞❡✜♥✐t✐♦♥✱ t❤❡ ❡q✉❛t✐♦♥ (∗ ∗ ∗) ✐s t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ s♣❤❡r❡✳

❋♦r♠ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ❝✐r❝✉❧❛r ❝②❧✐♥❞❡r ✇✐t❤ t❤❡ ❛①✐s Oz ❛♥❞ r❛❞✐✉s R✭❋✐❣✳ ✶✽✳✺✮✳ ▲❡t ✉s t❛❦❡ t❤❡ ❝♦♦r❞✐♥❛t❡ z(v) ❛♥❞ t❤❡ ❛♥❣❧❡ (u) ❢♦r♠❡❞ ❜② t❤❡♣❧❛♥❡ ♣❛ss✐♥❣ t❤r♦✉❣❤ t❤❡ z✲❛①✐s ❛♥❞ t❤❡ ♣♦✐♥t (x, y, z) ✇✐t❤ t❤❡ xz✲♣❧❛♥❡ ❛st❤❡ ♣❛r❛♠❡t❡rs u✱ v✱ ❝❤❛r❛❝t❡r✐③✐♥❣ t❤❡ ♣♦s✐t✐♦♥ ♦❢ t❤❡ ♣♦✐♥t (x, y, z) ♦♥ t❤❡❝②❧✐♥❞❡r✳ ❲❡ t❤❡♥ ❣❡t

x = R cosu, y = R sin u, z = v,

✇❤✐❝❤ ✐s t❤❡ r❡q✉✐r❡❞ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❝②❧✐♥❞❡r ✐♥ ♣❛r❛♠❡tr✐❝ ❢♦r♠✳❙q✉❛r✐♥❣ t❤❡ ✜rst t✇♦ ❡q✉❛t✐♦♥s ❛♥❞ ❛❞❞✐♥❣ t❡r♠✇✐s❡✱ ✇❡ ❣❡t t❤❡ ❡q✉❛t✐♦♥

♦❢ t❤❡ ❝②❧✐♥❞❡r ✐♥ ✐♠♣❧✐❝✐t ❢♦r♠✿

x2 + y2 = R2.

✶✽✳✺✳ ❊◗❯❆❚■❖◆❙ ❖❋ ❆ ❙❯❘❋❆❈❊ ❆◆❉ ❆ ❈❯❘❱❊ ■◆ ❙P❆❈❊ ✷✶✸

❋✐❣✉r❡ ✶✽✳✺✿ ❊①❡r❝✐s❡ ✸

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❝✉r✈❡ ✐♥ s♣❛❝❡✳ ❚❤❡ s②st❡♠ ♦❢ ❡q✉❛t✐♦♥s

f1(x, y, z) = 0, f2(x, y, z) = 0

✐s ❝❛❧❧❡❞ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ❝✉r✈❡ ✐♥ ✐♠♣❧✐❝✐t ❢♦r♠ ✐❢ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ❡❛❝❤♣♦✐♥t ♦❢ t❤❡ ❝✉r✈❡ s❛t✐s❢② ❜♦t❤ ❡q✉❛t✐♦♥s✳ ❆♥❞ ❝♦♥✈❡rs❡❧②✱ ❛♥② t❤r❡❡ ♥✉♠❜❡rss❛t✐s❢②✐♥❣ ❜♦t❤ ❡q✉❛t✐♦♥s r❡♣r❡s❡♥t t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ s♦♠❡ ♣♦✐♥t ♦♥ t❤❡❝✉r✈❡✳

❆ s②st❡♠ ♦❢ ❡q✉❛t✐♦♥s

x = ϕ1(t), y = ϕ2(t), z = ϕ3(t),

s♣❡❝✐❢②✐♥❣ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ♣♦✐♥ts ♦❢ t❤❡ ❝✉r✈❡ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ s♦♠❡ ♣❛✲r❛♠❡t❡r (t) ✐s t❡r♠❡❞ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ❝✉r✈❡ ✐♥ ♣❛r❛♠❡tr✐❝ ❢♦r♠✳

❚✇♦ s✉r❢❛❝❡s ✐♥t❡rs❡❝t✱ ❛s ❛ r✉❧❡✱ ❛❧♦♥❣ ❛ ❝✉r✈❡✳ ❖❜✈✐♦✉s❧②✱ ✐❢ t❤❡ s✉r❢❛❝❡s❛r❡ s♣❡❝✐✜❡❞ ❜② ❡q✉❛t✐♦♥s f1(x, y, z) = 0 ❛♥❞ f2(x, y, z) = 0✱ t❤❡♥ t❤❡ ❝✉r✈❡❛❧♦♥❣ ✇❤✐❝❤ t❤❡② ✐♥t❡rs❡❝t ✐s r❡♣r❡s❡♥t❡❞ ❜② ❛ s②st❡♠ ♦❢ ❡q✉❛t✐♦♥s

f1(x, y, z) = 0, f2(x, y, z) = 0.

❋♦r♠ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛♥ ❛r❜✐tr❛r② ❝✐r❝❧❡ ✐s s♣❛❝❡✳ ❆♥② ❝✐r❝❧❡ ❝❛♥ ❜❡ r❡♣✲r❡s❡♥t❡❞ ❛s ❛♥ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t✇♦ s♣❤❡r❡s✳ ❈♦♥s❡q✉❡♥t❧②✱ ❛♥② ❝✐r❝❧❡ ❝❛♥ ❜❡s♣❡❝✐✜❡❞ ❜② ❛ s②st❡♠ ♦❢ ❡q✉❛t✐♦♥s

(x− a1)2 + (y − b1)

2 + (z − c1)2 −R2

1 = 0,

(x− a2)2 + (y − b2)

2 + (z − c2)2 −R2

2 = 0.

}

❆s ❛ r✉❧❡✱ ❛ ❝✉r✈❡ ❛♥❞ ❛ s✉r❢❛❝❡ ✐♥t❡rs❡❝t ❛t s❡♣❛r❛t❡ ♣♦✐♥ts✳ ■❢ t❤❡ s✉r❢❛❝❡✐s s♣❡❝✐✜❡❞ ❜② t❤❡ ❡q✉❛t✐♦♥ f(x, y, z) = 0✱ ❛♥❞ t❤❡ ❝✉r✈❡ ❜② t❤❡ ❡q✉❛t✐♦♥s

✷✶✹❈❍❆P❚❊❘ ✶✽✳ ❘❊❈❚❆◆●❯▲❆❘ ❈❆❘❚❊❙■❆◆ ❈❖❖❘❉■◆❆❚❊❙ ■◆ ❙P❆❈❊

f1(x, y, z) = 0 ❛♥❞ f2(x, y, z) = 0✱ t❤❡♥ t❤❡ ♣♦✐♥ts ♦❢ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡ ❝✉r✈❡❛♥❞ t❤❡ s✉r❢❛❝❡ s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠ ♦❢ ❡q✉❛t✐♦♥s✿

f(x, y, z) = 0, f1 = (x, y, z) = 0, f2(x, y, z) = 0.

❙♦❧✈✐♥❣ t❤✐s s②st❡♠✱ ✇❡ ✜♥❞ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ♣♦✐♥ts ♦❢ ✐♥t❡rs❡❝t✐♦♥✳

❈❤❛♣t❡r ✶✾

❆ P❧❛♥❡ ❛♥❞ ❛ ❙tr❛✐❣❤t ▲✐♥❡

✶✾✳✶ ❚❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ♣❧❛♥❡

❋♦r♠ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛♥ ❛r❜✐tr❛r② ♣❧❛♥❡ ✐♥ t❤❡ r❡❝t❛♥❣✉❧❛r ❈❛rt❡s✐❛♥ ❝♦♦r❞✐✲♥❛t❡s xyz✳

▲❡t A0(x0, y0, z0) ❜❡ ❛ ♣♦✐♥t ✐♥ ❛ ♣❧❛♥❡ ❛♥❞ n ❛ ♥♦♥③❡r♦ ✈❡❝t♦r ♣❡r♣❡♥✲❞✐❝✉❧❛r t♦ t❤❡ ♣❧❛♥❡✳ ❚❤❡♥ ✇❤❛t❡✈❡r t❤❡ ♣♦✐♥t ♦❢ t❤❡ ♣❧❛♥❡ A(x, y, z) ✐s✱ t❤❡✈❡❝t♦rs

−−→A0A ❛♥❞ n ❛r❡ ♠✉t✉❛❧❧② ♣❡r♣❡♥❞✐❝✉❧❛r ✭❋✐❣✳ ✶✾✳✶✮✳ ❍❡♥❝❡✱

−−→A0A · n = 0. (∗)

▲❡t a✱ b✱ c ❜❡ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ✈❡❝t♦r n ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❜❛s✐s exey✱ ez✳

❚❤❡♥✱ s✐♥❝❡−−→A0A =

−→OA−−−→

OA0✱ ✐t ❢♦❧❧♦✇s ❢r♦♠ (∗)

a(x− x0) + b(y − y0) + c(z − z0) = 0. (∗∗)

❋✐❣✉r❡ ✶✾✳✶✿ ❊q✉❛t✐♦♥ ♦❢ ❛ ♣❧❛♥❡

✷✶✺

✷✶✻ ❈❍❆P❚❊❘ ✶✾✳ ❆ P▲❆◆❊ ❆◆❉ ❆ ❙❚❘❆■●❍❚ ▲■◆❊

❚❤✐s ✐s t❤❡ r❡q✉✐r❡❞ ❡q✉❛t✐♦♥✳❚❤✉s✱ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛♥② ♣❧❛♥❡ ✐s ❧✐♥❡❛r r❡❧❛t✐✈❡ t♦ t❤❡ ❝♦♦r❞✐♥❛t❡s x✱

y✱ z✳❙✐♥❝❡ t❤❡ ❢♦r♠✉❧❛s ❢♦r tr❛♥s✐t✐♦♥ ❢r♦♠ ♦♥❡ ❈❛rt❡s✐❛♥ s②st❡♠ ♦❢ ❝♦♦r❞✐♥❛t❡s

t♦ ❛♥♦t❤❡r ❛r❡ ❧✐♥❡❛r✱ ✇❡ ♠❛② st❛t❡ t❤❛t t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ♣❧❛♥❡ ✐s ❧✐♥❡❛r ✐♥❛♥② ❈❛rt❡s✐❛♥ s②st❡♠ ♦❢ ❝♦♦r❞✐♥❛t❡s ✭❜✉t ♥♦t ♦♥❧② ✐♥ ❛ r❡❝t❛♥❣✉❧❛r ♦♥❡✮✳

▲❡t ✉s ♥♦✇ s❤♦✇ t❤❛t ❛♥② ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❢♦r♠

ax+ by + cz + d = 0

✐s t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ♣❧❛♥❡✳▲❡t x0✱ y0✱ z0 ❜❡ ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❣✐✈❡♥ ❡q✉❛t✐♦♥✳ ❚❤❡♥

ax0 + by0 + cz0 + d = 0

❛♥❞ t❤❡ ❡q✉❛t✐♦♥ ♠❛② ❜❡ r❡✇r✐tt❡♥ ✐♥ t❤❡ ❢r♦♠

a(x− x0) + b(y − y0) + c(z − z0) = 0. (∗ ∗ ∗)

▲❡t n ❜❡ ❛ ✈❡❝t♦r ✇✐t❤ t❤❡ ❝♦♦r❞✐♥❛t❡s a✱ b✱ c ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❜❛s✐sex✱ ey✱ ez✱ A0 ❛ ♣♦✐♥t ✇✐t❤ t❤❡ ❝♦♦r❞✐♥❛t❡s x0✱ y0✱ z0 ❛♥❞ A ❛ ♣♦✐♥t ✇✐t❤ t❤❡❝♦♦r❞✐♥❛t❡s x✱ y✱ z✳ ❚❤❡♥ t❤❡ ❡q✉❛t✐♦♥ (∗∗∗) ❝❛♥ ❜❡ ✇r✐tt❡♥ ✐♥ t❤❡ ❡q✉✐✈❛❧❡♥t❢♦r♠ −−→

A0A · n = 0.

❲❤❡♥❝❡ ✐t ❢♦❧❧♦✇s t❤❛t ❛❧❧ ♣♦✐♥ts ♦❢ t❤❡ ♣❧❛♥❡ ♣❛ss✐♥❣ t❤r♦✉❣❤ t❤❡ ♣♦✐♥t A0

❛♥❞ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ✈❡❝t♦r n ✭❛♥❞ ♦♥❧② t❤❡②✮ s❛t✐s❢② t❤❡ ❣✐✈❡♥ ❡q✉❛t✐♦♥❛♥❞✱ ❝♦♥s❡q✉❡♥t❧②✱ ✐t ✐s t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤✐s ♣❧❛♥❡✳

▲❡t ✉s ♥♦t❡ t❤❛t t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ x✱ y✱ z ✐♥ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ♣❧❛♥❡ ❛r❡t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ✈❡❝t♦r ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ♣❧❛♥❡ r❡❧❛t✐✈❡ t♦ t❤❡ ❜❛s✐sex ey✱ ez✳

✶✾✳✷ ❊①❡r❝✐s❡s

✶✳ ❋♦r♠ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ♣❧❛♥❡ ❣✐✈❡♥ t✇♦ ♣♦✐♥ts (x1, y1, z1) ❛♥❞ (x2, y2, z2)s✐t✉❛t❡❞ s②♠♠❡tr✐❝❛❧❧② ❛❜♦✉t ✐t✳

✷✳ ❙❤♦✇ t❤❛t t❤❡ ♣❧❛♥❡s

ax+ by + cz + d1 = 0,

ax+ by + cz + d2 = 0, d1 6= d2,

✶✾✳✷✳ ❊❳❊❘❈■❙❊❙ ✷✶✼

❛r❡ ♣❛r❛❧❧❡❧ ✭❞♦ ♥♦t ✐♥t❡rs❡❝t✮✳✸✳ ❲❤❛t ✐s t❤❡ ❧♦❝✉s ♦❢ ♣♦✐♥ts ✇❤♦s❡ ❝♦♦r❞✐♥❛t❡s s❛t✐s❢② t❤❡ ❡q✉❛t✐♦♥

(ax+ by + cz + d)2 − (αx+ βy + γz + δ)2 = 0?

✹✳ ❙❤♦✇ t❤❛t t❤❡ ❝✉r✈❡ r❡♣r❡s❡♥t❡❞ ❜② t❤❡ ❡q✉❛t✐♦♥s

f(x, y, z) + a1x+ b1y + c1z + d1 = 0,

f(x, y, z) + a2x+ b2y + c2z + d1 = 0,

✐s ❛ ♣❧❛♥❡ ♦♥❡✱ ✐✳❡✳ ❛❧❧ ♣♦✐♥ts ♦❢ t❤✐s ❝✉r✈❡ ❜❡❧♦♥❣ t♦ ❛ ♣❧❛♥❡✳✺✳ ❙❤♦✇ t❤❛t t❤❡ t❤r❡❡ ♣❧❛♥❡s s♣❡❝✐✜❡❞ ❜② t❤❡ ❡q✉❛t✐♦♥s

ax+ by + cz + d = 0,

αx+ βy + γz + d = 0,

λ(ax+ by + cz) + µ(alphax+ βy + γz) + k = 0,

❤❛✈❡ ♥♦ ♣♦✐♥ts ✐♥ ❝♦♠♠♦♥ ✐❢ k 6= λd+ µδ✳✻✳ ❲r✐t❡ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ♣❧❛♥❡ ♣❛ss✐♥❣ t❤r♦✉❣❤ t❤❡ ❝✐r❝❧❡ ♦❢ ✐♥t❡rs❡❝✲

t✐♦♥ ♦❢ t❤❡ t✇♦ s♣❤❡r❡s

x2 + y2 + z2 + ax+ by + cz + d = 0,

x2 + y2 + z2 + αx+ βy + γz + δ = 0.

✼✳ ❙❤♦✇ t❤❛t ✐♥✈❡rs✐♦♥ tr❛♥s❢♦r♠s ❛ s♣❤❡r❡ ❡✐t❤❡r ✐♥t♦ ❛ s♣❤❡r❡ ♦r ✐♥t♦ ❛♣❧❛♥❡✳

✽✳ ❙❤♦✇ t❤❛t t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛♥② ♣❧❛♥❡ ♣❛ss✐♥❣ t❤r♦✉❣❤ t❤❡ ❧✐♥❡ ♦❢ ✐♥t❡r✲s❡❝t✐♦♥ ♦❢ t❤❡ ♣❧❛♥❡s

ax+ by + cz + d = 0,

αx+ βy + γz + δ = 0,

❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ✐♥ t❤❡ ❢r♦♠

λ(ax+ by + cz + d) + µ(αx+ βy + γz + δ) = 0.

✾✳ ❙❤♦✇ t❤❛t t❤❡ ♣❧❛♥❡ ♣❛ss✐♥❣ t❤r♦✉❣❤ t❤❡ t❤r❡❡ ❣✐✈❡♥ ♣♦✐♥ts (xi, yi, zi)(i = 1, 2, 3) ✐s s♣❡❝✐✜❡❞ ❜② t❤❡ ❡q✉❛t✐♦♥

∣

∣

∣

∣

∣

∣

∣

∣

x y z 1x1 y1 z1 1x2 y2 z2 1x3 y3 z3 1

∣

∣

∣

∣

∣

∣

∣

∣

= 0.

✷✶✽ ❈❍❆P❚❊❘ ✶✾✳ ❆ P▲❆◆❊ ❆◆❉ ❆ ❙❚❘❆■●❍❚ ▲■◆❊

✶✾✳✸ ❙♣❡❝✐❛❧ ♣♦s✐t✐♦♥s ♦❢ ❛ ♣❧❛♥❡ r❡❧❛t✐✈❡ t♦ ❝♦✲

♦r❞✐♥❛t❡ s②st❡♠

▲❡t ✉s ✜♥❞ ♦✉t t❤❡ ♣❡❝✉❧✐❛r✐t✐❡s ♦❢ t❤❡ ♦❢ t❤❡ ♣♦s✐t✐♦♥ ♦❢ ❛ ♣❧❛♥❡ r❡❧❛t✐✈❡t♦ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ✇❤✐❝❤ t❛❦❡ ♣❧❛❝❡ ✇❤❡♥ ✐ts ❡q✉❛t✐♦♥ ✐s ♦❢ t❤✐s ♦r t❤❛t♣❛rt✐❝✉❧❛r ❢♦r♠✳

✶✳ a = 0✱ b = 0✳ ❱❡❝t♦r n ✭♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ♣❧❛♥❡✮ ✐s ♣❛r❛❧❧❡❧ t♦ t❤❡z✲❛①✐s✳ ❚❤❡ ♣❧❛♥❡ ✐s ♣❛r❛❧❧❡❧ t♦ t❤❡ xy✲♣❧❛♥❡✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✐t ❝♦✐♥❝✐❞❡s✇✐t❤ t❤❡ xy✲♣❧❛♥❡ ✐❢ d ✐s ❛❧s♦ ③❡r♦✳

✷✳ b = 0✱ c = 0✳ ❚❤❡ ♣❧❛♥❡ ✐s ♣❛r❛❧❧❡❧ t♦ t❤❡ yz✲♣❧❛♥❡ ❛♥❞ ❝♦✐♥❝✐❞❡s ✇✐t❤✐t ✐❢ d = 0✳

✸✳ c = 0✱ a = 0✳ ❚❤❡ ♣❧❛♥❡ ✐s ♣❛r❛❧❧❡❧ t♦ t❤❡ xz✲♣❧❛♥❡ ❛♥❞ ❝♦✐♥❝✐❞❡s ✇✐t❤✐t ✐❢ d = 0✳

✹✳ a = 0✱ b 6= 0✱ c 6= 0✳ ❱❡❝t♦r n ✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ x✲❛①✐s✿ exn = 0✳❚❤❡ ♣❧❛♥❡ ✐s ♣❛r❛❧❧❡❧ t♦ t❤❡ x✲❛①✐s✱ ✐♥ ♣❛rt✐❝✉❧❛r✱ ✐t ♣❛ss❡s t❤r♦✉❣❤ ✐t ✐❢d = 0✳

✺✳ a 6= 0✱ b = 0✱ c 6= 0✳ ❚❤❡ ♣❧❛♥❡ ✐s ♣❛r❛❧❧❡❧ t♦ t❤❡ y✲❛①✐s ❛♥❞ ♣❛ss❡st❤r♦✉❣❤ ✐t ✐❢ d = 0✳

✻✳ a 6= 0✱ b 6= 0✱ c = 0✳ ❚❤❡ ♣❧❛♥❡ ✐s ♣❛r❛❧❧❡❧ t♦ t❤❡ z✲❛①✐s ❛♥❞ ♣❛ss❡st❤r♦✉❣❤ ✐t ✐❢ d = 0✳

✼✳ d = 0✳ ❚❤❡ ♣❧❛♥❡ ♣❛ss❡s t❤r♦✉❣❤ t❤❡ ♦r✐❣✐♥ ✭✇❤♦s❡ ❝♦♦r❞✐♥❛t❡s ✵✱ ✵✱ ✵s❛t✐s❢② t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ♣❧❛♥❡✮✳

■❢ ❛❧❧ t❤❡ ❝♦❡✣❝✐❡♥ts ❛r❡ ♥♦♥✲③❡r♦✱ t❤❡♥ t❤❡ ❡q✉❛t✐♦♥ ♠❛② ❜❡ ❞✐✈✐❞❡❞ ❜②−d✳ ❚❤❡♥✱ ♣✉tt✐♥❣

−d

a= α, −d

b= β, −d

c= γ,

✇❡ ❣❡t t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ♣❧❛♥❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✿

x

α+

y

β+

z

γ= 1. (∗)

❚❤❡ ♥✉♠❜❡rs α✱ β✱ γ ❛r❡ ❡q✉❛❧ ✭✉♣ t♦ ❛ s✐❣♥✮ t♦ t❤❡ s❡❣♠❡♥ts ✐♥t❡r❝❡♣t❡❞❜② t❤❡ ♣❧❛♥❡ ♦♥ t❤❡ ❝♦♦r❞✐♥❛t❡ ❛①❡s✳ ■♥❞❡❡❞✱ t❤❡ x✲❛①✐s ✭y = 0✱ z = 0✮ ✐s✐♥t❡rs❡❝t❡❞ ❜② t❤❡ ♣❧❛♥❡ ❛t ♣♦✐♥t (α, 0, 0)✱ t❤❡ y✲❛①✐s ❛t ♣♦✐♥t (0, β, 0)✱ ❛♥❞

✶✾✳✹✳ ❊❳❊❘❈■❙❊❙ ✷✶✾

t❤❡ z✲❛①✐s ❛t ♣♦✐♥t (0, 0, γ)✳ ❚❤❡ ❡q✉❛t✐♦♥ (∗) ✐s ❝❛❧❧❡❞ t❤❡ ✐♥t❡r❝❡♣t ❢♦r♠ ♦❢t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ♣❧❛♥❡✳

❲❡ ❝♦♥❝❧✉❞❡ ✇✐t❤ ❛ ♥♦t❡ t❤❛t ❛♥② ♣❧❛♥❡ ♥♦t ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ xy✲♣❧❛♥❡(c 6= 0) ♠❛② ❜❡ s♣❡❝✐✜❡❞ ❜② ❛♥ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❢♦r♠

z = px+ qy + l.

✶✾✳✹ ❊①❡r❝✐s❡s

✶✳ ❋✐♥❞ t❤❡ ❝♦♥❞✐t✐♦♥s ✉♥❞❡r ✇❤✐❝❤ t❤❡ ♣❧❛♥❡

ax+ by + cz + d = 0

✐♥t❡rs❡❝ts t❤❡ ♣♦s✐t✐✈❡ s❡♠✐✲❛①✐s x(y, z)✳✷✳ ❋✐♥❞ t❤❡ ✈♦❧✉♠❡ ♦❢ t❤❡ t❡tr❛❤❡❞r♦♥ ❜♦✉♥❞❡❞ ❜② t❤❡ ❝♦♦r❞✐♥❛t❡ ♣❧❛♥❡s

❛♥❞ t❤❡ ♣❧❛♥❡ax+ by + cz + d = 0

✐❢ abcd 6= 0✳✸✳ Pr♦✈❡ t❤❛t t❤❡ ♣♦✐♥ts ✐♥ s♣❛❝❡ ❢♦r ✇❤✐❝❤

|x|+ |y|+ |z| < a,

❛r❡ s✐t✉❛t❡❞ ✐♥s✐❞❡ ❛♥ ♦❝t❛❤❡❞r♦♥ ✇✐t❤ ❝❡♥tr❡ ❛t t❤❡ ♦r✐❣✐♥ ❛♥❞ t❤❡ ♦♥ t❤❡✈❡rt✐❝❡s ❝♦♦r❞✐♥❛t❡ ❛①❡s✳

✹✳ ●✐✈❡♥ ❛ ♣❧❛♥❡ σ ❜② t❤❡ ❡q✉❛t✐♦♥ ✐♥ r❡❝t❛♥❣✉❧❛r ❈❛rt❡s✐❛♥ ❝♦♦r❞✐♥❛t❡s

ax+ by + cz + d = 0.

❋♦r♠ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ♣❧❛♥❡ σ′ s②♠♠❡tr✐❝❛❧ t♦ σ ❛❜♦✉t t❤❡ xy✲♣❧❛♥❡✭❛❜♦✉t t❤❡ ♦r✐❣✐♥ O✮✳

✺✳ ●✐✈❡♥ ❛ ❢❛♠✐❧② ♦❢ ♣❧❛♥❡s ❞❡♣❡♥❞✐♥❣ ♦♥ ❛ ♣❛r❛♠❡t❡r

ax+ by + cz + d+ λ(αx+ βy + γz + δ) = 0.

❋✐♥❞ ✐♥ t❤✐s ❢❛♠✐❧② ❛ ♣❧❛♥❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ z✲❛①✐s✳✼✳ ■♥ t❤❡ ❢❛♠✐❧② ♦❢ ♣❧❛♥❡s

(a1x+ b1y + c1z + d1) + λ(a2x+ b2y + c2z + d2)

+ µ(a3x+ b3y + c3z + d3) = 0

✜♥❞ t❤❡ ♣❧❛♥❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ xy✲♣❧❛♥❡✳ ❚❤❡ ♣❛r❛♠❡t❡rs ♦❢ t❤❡ ❢❛♠✐❧② ❛r❡ λ❛♥❞ µ✳

✷✷✵ ❈❍❆P❚❊❘ ✶✾✳ ❆ P▲❆◆❊ ❆◆❉ ❆ ❙❚❘❆■●❍❚ ▲■◆❊

✶✾✳✺ ❚❤❡ ♥♦r♠❛❧ ❢♦r♠ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ♣❧❛♥❡

■❢ ❛ ♣♦✐♥t A(x, y, z) ❜❡❧♦♥❣s t♦ t❤❡ ♣❧❛♥❡

ax+ by + cz + d = 0, (∗)

t❤❡♥ ✐ts ❝♦♦r❞✐♥❛t❡s s❛t✐s❢② t❤❡ ❡q✉❛t✐♦♥ (∗)✳▲❡t ✉s ✜♥❞ ♦✉t ✇❤❛t ❣❡♦♠❡tr✐❝❛❧ ♠❡❛♥✐♥❣ ❤❛s t❤❡ ❡①♣r❡ss✐♦♥

ax+ by + cz + d

✐❢ t❤❡ ♣♦✐♥t A ❞♦❡s ♥♦t ❜❡❧♦♥❣ t♦ t❤❡ ♣❧❛♥❡✳❲❡ ❞r♦♣ ❢r♦♠ t❤❡ ♣♦✐♥tA ❛ ♣❡r♣❡♥❞✐❝✉❧❛r ♦♥t♦ t❤❡ ♣❧❛♥❡✳ ▲❡tA0(x0, y0, z0)

❜❡ t❤❡ ❢♦♦t ♦❢ t❤❡ ♣❡r♣❡♥❞✐❝✉❧❛r✳ ❙✐♥❝❡ t❤❡ ♣♦✐♥t A0 ❧✐❡s ♦♥ t❤❡ ♣❧❛♥❡✱ t❤❡♥

ax0 + by0 + cz0 + d = 0.

❲❤❡♥❝❡

ax+ by + cz + d = a(x− x0) + b(y − y0) + c(z − z0) = n · −−→A0A = ±|n|δ,

✇❤❡r❡ n ✐s ❛ ✈❡❝t♦r ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ♣❧❛♥❡✱ ✇✐t❤ t❤❡ ❝♦♦r❞✐♥❛t❡s a✱ b✱ c✱❛♥❞ δ ✐s t❤❡ ❞✐st❛♥❝❡ ♦❢ t❤❡ ♣♦✐♥t A ❢♦r♠ t❤❡ ♣❧❛♥❡✳

❚❤✉sax+ by + cz + d

✐s ♣♦s✐t✐✈❡ ♦♥ ♦♥❡ s✐❞❡ ♦❢ t❤❡ ♣❧❛♥❡✱ ❛♥❞ ♥❡❣❛t✐✈❡ ♦♥ t❤❡ ♦t❤❡r✱ ✐ts ❛❜s♦❧✉t❡✈❛❧✉❡ ❜❡✐♥❣ ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❞✐st❛♥❝❡ ♦❢ t❤❡ ♣♦✐♥t A ❢r♦♠ t❤❡ ♣❧❛♥❡✳ ❚❤❡♣r♦♣♦rt✐♦♥❛❧✐t② ❢❛❝t♦r

±|n| =√a2 + b2 + c2.

■❢ ✐♥ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ♣❧❛♥❡

a2 + b2 + c2 = 1,

t❤❡♥ax+ by + cz + d,

✇✐❧❧ ❜❡ ❡q✉❛❧ ✉♣ t♦ ❛ s✐❣♥ t♦ t❤❡ ❞✐st❛♥❝❡ ♦❢ t❤❡ ♣♦✐♥t ❢r♦♠ t❤❡ ♣❧❛♥❡✳ ■♥ t❤✐s❝❛s❡ t❤❡ ♣❧❛♥❡ ✐s s❛✐❞ t♦ ❜❡ s♣❡❝✐✜❡❞ ❜② ❛♥ ❡q✉❛t✐♦♥ ✐♥ t❤❡ ♥♦r♠❛❧ ❢♦r♠✳

❖❜✈✐♦✉s❧②✱ t♦ ♦❜t❛✐♥ t❤❡ ♥♦r♠❛❧ ❢♦r♠ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ♣❧❛♥❡ (∗)✱ ✐t ✐ss✉✣❝✐❡♥t t♦ ❞✐✈✐❞❡ ✐t ❜②

±√a2 + b2 + c2.

✶✾✳✻✳ ❊❳❊❘❈■❙❊❙ ✷✷✶

✶✾✳✻ ❊①❡r❝✐s❡s

✶✳ ❚❤❡ ♣❧❛♥❡s s♣❡❝✐✜❡❞ ❜② t❤❡ ❡q✉❛t✐♦♥s ✐♥ r❡❝t❛♥❣✉❧❛r ❈❛rt❡s✐❛♥ ❝♦♦r✲❞✐♥❛t❡s

ax+ by + cz + d = 0,

ax+ by + cz + d′ = 0

✇❤❡r❡ d 6= d′✱ ❤❛✈❡ ♥♦ ♣♦✐♥ts ✐♥ ❝♦♠♠♦♥✱ ❤❡♥❝❡✱ t❤❡② ❛r❡ ♣❛r❛❧❧❡❧✳ ❋✐♥❞ t❤❡❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡s❡ ♣❧❛♥❡s✳

✷✳ ❚❤❡ ♣❧❛♥❡ax+ by + cz + d = 0

✐s ♣❛r❛❧❧❡❧ t♦ z✲❛①✐s✳ ❋✐♥❞ t❤❡ ❞✐st❛♥❝❡ ♦❢ t❤❡ z✲❛①✐s ❢r♦♠ t❤✐s ♣❧❛♥❡✳✸✳ ❲❤❛t ✐s t❤❡ ❧♦❝✉s ♦❢ ♣♦✐♥ts ✇❤♦s❡ ❞✐st❛♥❝❡ t♦ t✇♦ ❣✐✈❡♥ ♣❧❛♥❡s ❛r❡ ✐♥

❛ ❣✐✈❡♥ r❛t✐♦❄✹✳ ❋♦r♠ t❤❡ ❡q✉❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡s ♣❛r❛❧❧❡❧ t♦ t❤❡ ♣❧❛♥❡

ax+ by + cz + d = 0

❛♥❞ ❧♦❝❛t❡❞ ❛t ❛ ❞✐st❛♥❝❡ δ ❢r♦♠ ✐t✳✺✳ ❙❤♦✇ t❤❛t t❤❡ ♣♦✐♥ts ✐♥ s♣❛❝❡ s❛t✐s❢②✐♥❣ t❤❡ ❝♦♥❞✐t✐♦♥

|ax+ by + cz + d| < δ2,

❛r❡ s✐t✉❛t❡❞ ❜❡t✇❡❡♥ t❤❡ ♣❛r❛❧❧❡❧ ♣❧❛♥❡s

ax+ by + cz + d± δ2 = 0.

✻✳ ●✐✈❡♥ ❛r❡ t❤❡ ❡q✉❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡s ❝♦♥t❛✐♥✐♥❣ t❤❡ ❢❛❝❡s ♦❢ ❛ t❡tr❛✲❤❡❞r♦♥ ❛♥❞ ❛ ♣♦✐♥t M ❜② ✐ts ❝♦♦r❞✐♥❛t❡s✳ ❍♦✇ t♦ ✜♥❞ ♦✉t ✇❤❡t❤❡r ♦r ♥♦tt❤❡ ♣♦✐♥t M ❧✐❡s ✐♥s✐❞❡ t❤❡ t❡tr❛❤❡❞r♦♥❄

✼✳ ❉❡r✐✈❡ t❤❡ ❢♦r♠✉❧❛s ❢♦r tr❛♥s✐t✐♦♥ t♦ ❛ ♥❡✇ s②st❡♠ ♦❢ r❡❝t❛♥❣✉❧❛r❈❛rt❡s✐❛♥ ❝♦♦r❞✐♥❛t❡s x′y′z′ ✐❢ t❤❡ ♥❡✇ ❝♦♦r❞✐♥❛t❡ ♣❧❛♥❡ ❛r❡ s♣❡❝✐✜❡❞ ✐♥ t❤❡♦❧❞ s②st❡♠ ❜② t❤❡ ❡q✉❛t✐♦♥s

a1x+ b1y + c1z + d1 = 0,

a2x+ b2y + c2z + d2 = 0,

a3x+ b3y + c3z + d3 = 0.

✷✷✷ ❈❍❆P❚❊❘ ✶✾✳ ❆ P▲❆◆❊ ❆◆❉ ❆ ❙❚❘❆■●❍❚ ▲■◆❊

✶✾✳✼ ❘❡❧❛t✐✈❡ ♣♦s✐t✐♦♥ ♦❢ ♣❧❛♥❡s

❙✉♣♣♦s❡ ✇❡ t✇♦ ♣❧❛♥❡s

a1x+ b1y + c1z + d1 = 0,

a2x+ b2y + c2z + d2 = 0.

}

(∗)

❋✐♥❞ ♦✉t ✉♥❞❡r ✇❤✐❝❤ ❝♦♥❞✐t✐♦♥ t❤❡s❡ ♣❧❛♥❡s ❛r❡✿ ✭❛✮ ♣❛r❛❧❧❡❧✱ ✭❜✮♠✉t✉❛❧❧②♣❡r♣❡♥❞✐❝✉❧❛r✳

❙✐♥❝❡ a1✱ b1✱ c1 ❛r❡ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ✈❡❝t♦r n1 ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ✜rst♣❧❛♥❡✱ ❛♥❞ a2✱ b2✱ c2 ❛r❡ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ✈❡❝t♦r n2 ✇❤✐❝❤ ✐s ♣❡r♣❡♥❞✐❝✉❧❛rt♦ t❤❡ s❡❝♦♥❞ ♣❧❛♥❡✱ t❤❡ ♣❧❛♥❡s ❛r❡ ♣❛r❛❧❧❡❧ ✐❢ t❤❡ ✈❡❝t♦rs n1✱ n2 ❛r❡ ♣❛r❛❧❧❡❧✱✐✳❡✳ ✐❢ t❤❡✐r ❝♦♦r❞✐♥❛t❡s ❛r❡ ♣r♦♣♦rt✐♦♥❛❧✿

a1a2

=b1b2

=c1c2.

▼♦r❡♦✈❡r✱ t❤✐s ❝♦♥❞✐t✐♦♥ ✐s s✉✣❝✐❡♥t ❢♦r ♣❛r❛❧❧❡❧✐s♠ ♦❢ t❤❡ ♣❧❛♥❡s ✐❢ t❤❡② ❛r❡♥♦t ❝♦✐♥❝✐❞❡♥t✳

❋♦r t❤❡ ♣❧❛♥❡s (∗) t♦ ❜❡ ♠✉t✉❛❧❧② ♣❡r♣❡♥❞✐❝✉❧❛r ✐t ✐s ♥❡❝❡ss❛r② ❛♥❞ s✉❢✲✜❝✐❡♥t t❤❛t t❤❡ ♠❡♥t✐♦♥❡❞ ✈❡❝t♦rs n1 ❛♥❞ n2 ❛r❡ ♠✉t✉❛❧❧② ♣❡r♣❡♥❞✐❝✉❧❛r✱✇❤✐❝❤ ❢♦r ♥♦♥✲③❡r♦ ✈❡❝t♦rs ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ❝♦♥❞✐t✐♦♥

n1n2 = 0 ♦r a1a2 + b1b2 + c1c2 = 0.

▲❡t t❤❡ ❡q✉❛t✐♦♥s (∗) s♣❡❝✐❢② t✇♦ ❛r❜✐tr❛r② ♣❧❛♥❡s✳ ❋✐♥❞ t❤❡ ❛♥❣❧❡ ♠❛❞❡❜② t❤❡s❡ ♣❧❛♥❡s✳

❚❤❡ ❛♥❣❧❡ θ ❜❡t✇❡❡♥ t❤❡ ✈❡❝t♦rs n1 ❛♥❞ n2 ✐s ❡q✉❛❧ t♦ ♦♥❡ t❤❡ ❛♥❣❧❡s❢♦r♠❡❞ ❜② ♣❧❛♥❡s ❛♥❞ ✐s r❡❛❞✐❧② ❢♦✉♥❞✳ ❲❡ ❤❛✈❡

n1 · n2 = |n1| |n2| cos θ.

❲❤❡♥❝❡

cos θ =a1a2 + b1b2 + c1c2

√

a21 + b21 + c21√

a22 + b22 + c22.

✶✾✳✽ ❊q✉❛t✐♦♥s ♦❢ t❤❡ str❛✐❣❤t ❧✐♥❡

❆♥② str❛✐❣❤t ❧✐♥❡ ❝❛♥ ❜❡ s♣❡❝✐✜❡❞ ❛s ❛♥ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t✇♦ ♣❧❛♥❡s✳ ❈♦♥s❡✲q✉❡♥t❧②✱ ❛♥② str❛✐❣❤t ❧✐♥❡ ❝❛♥ ❜❡ s♣❡❝✐✜❡❞ ❜② t❤❡ ❡q✉❛t✐♦♥s

a1x+ b1y + c1z + d1 = 0,

a2x+ b2y + c2z + d2 = 0.

}

(∗)

✶✾✳✽✳ ❊◗❯❆❚■❖◆❙ ❖❋ ❚❍❊ ❙❚❘❆■●❍❚ ▲■◆❊ ✷✷✸

❋✐❣✉r❡ ✶✾✳✷✿ ❊q✉❛t✐♦♥s ♦❢ ❛ ❧✐♥❡ ✐ s♣❛❝❡

t❤❡ ✜rst ✇❤✐❝❤ r❡♣r❡s❡♥ts ♦♥❡ ♣❧❛♥❡ ❛♥❞ t❤❡ s❡❝♦♥❞ t❤❡ ♦t❤❡r✳ ❈♦♥✈❡rs❡❧②✱ ❛♥②❝♦♠♣❛t✐❜❧❡ s②st❡♠ ♦❢ t✇♦ s✉❝❤ ✐♥❞❡♣❡♥❞❡♥t ❡q✉❛t✐♦♥s r❡♣r❡s❡♥ts t❤❡ ❡q✉❛t✐♦♥s♦❢ ❛ str❛✐❣❤t ❧✐♥❡✳

▲❡t A0(x0, y0, z0) ❜❡ ✜①❡❞ ♣♦✐♥t ♦♥ ❛ str❛✐❣❤t ❧✐♥❡✱ A(x, y, z) ❛♥ ❛r❜✐✲tr❛r② ♣♦✐♥t ♦❢ t❤❡ str❛✐❣❤t ❧✐♥❡✱ ❛♥❞ e(k, l,m) ❛ ♥♦♥✲③❡r♦ ✈❡❝t♦r ♣❛r❛❧❧❡❧ t♦t❤❡ str❛✐❣❤t ❧✐♥❡ ✭❋✐❣✳ ✶✾✳✷✮✳ ❚❤❡♥ t❤❡ ✈❡❝t♦rs

−−→A0A ❛♥❞ e ❛r❡ ♣❛r❛❧❧❡❧ ❛♥❞✱

❝♦♥s❡q✉❡♥t❧②✱ t❤❡✐r ❝♦♦r❞✐♥❛t❡s ❛r❡ ♣r♦♣♦rt✐♦♥❛❧✱ ✐✳❡✳

x− x0

k=

y − y0l

=z − z0m

. (∗∗)

❚❤✐s ❢♦r♠ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ str❛✐❣❤t ❧✐♥❡ ✐s ❝❛❧❧❡❞ ❝❛♥♦♥✐❝❛❧✳ ■t r❡♣r❡✲s❡♥ts ❛ ♣❛rt✐❝✉❧❛r ❝❛s❡ (∗)✱ s✐♥❝❡ ✐t ❛❧❧♦✇s ❛♥ ❡q✉✐✈❛❧❡♥t ❢♦r♠

x− x0

k=

y − y0l

,y − y0

l=

z − z0m

,

❝♦rr❡s♣♦♥❞✐♥❣ t♦ (∗)✳❙✉♣♣♦s❡ ❛ str❛✐❣❤t ❧✐♥❡ ✐s r❡♣r❡s❡♥t❡❞ ❜② t❤❡ ❡q✉❛t✐♦♥s (∗)✳ ▲❡t ✉s ❢♦r♠

✐ts ❡q✉❛t✐♦♥ ✐♥ ❝❛♥♦♥✐❝❛❧ ❢♦r♠✳ ❋♦r t❤✐s ♣✉r♣♦s❡ ✐t ✐s s✉✣❝✐❡♥t t♦ ✜♥❞ ❛ ♣♦✐♥tA0 ♦♥ t❤❡ str❛✐❣❤t ❧✐♥❡ ❛♥❞ ❛ ✈❡❝t♦r e ♣❛r❛❧❧❡❧ t♦ t❤✐s ❧✐♥❡✳

❆♥② ✈❡❝t♦r e(k, l,m) ♣❛r❛❧❧❡❧ t♦ t❤❡ str❛✐❣❤t ❧✐♥❡ ✇✐❧❧ ❜❡ ♣❛r❛❧❧❡❧ t♦ ❡✐t❤❡r♦❢ t❤❡ ♣❧❛♥❡s (∗)✱ ❛♥❞ ❝♦♥✈❡rs❡❧②✳ ❈♦♥s❡q✉❡♥t❧②✱ k✱ l✱ m s❛t✐s❢② t❤❡ ❡q✉❛t✐♦♥s

a1k + b1l + c1m = 0,

a2k + b2l + c2m = 0.

}

(∗ ∗ ∗)

❚❤✉s✱ ❛♥② s♦❧✉t✐♦♥ ♦❢ t❤❡ s②st❡♠ (∗) ♠❛② ❜❡ t❛❦❡♥ ❛s x0✱ y0✱ z0 ❢♦r t❤❡❝❛♥♦♥✐❝❛❧ ❡q✉❛t✐♦♥ ♦❢ t❤❡ str❛✐❣❤t ❧✐♥❡ ❛♥❞ ❛♥② s♦❧✉t✐♦♥ ♦❢ (∗ ∗ ∗) ❛s t❤❡❝♦❡✣❝✐❡♥ts k✱ l✱ m✱ ❢♦r ✐♥st❛♥❝❡

k =

∣

∣

∣

∣

b1 c1b2 c2

∣

∣

∣

∣

, l =

∣

∣

∣

∣

c1 a1c2 a2

∣

∣

∣

∣

, m =

∣

∣

∣

∣

a1 b1a2 b2

∣

∣

∣

∣

.

✷✷✹ ❈❍❆P❚❊❘ ✶✾✳ ❆ P▲❆◆❊ ❆◆❉ ❆ ❙❚❘❆■●❍❚ ▲■◆❊

❋r♦♠ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ str❛✐❣❤t ❧✐♥❡ ✐♥ ❝❛♥♦♥✐❝❛❧ ❢♦r♠ ✇❡ ❝❛♥ ❞❡r✐✈❡ ✐ts❡q✉❛t✐♦♥s ✐♥ ♣❛r❛♠❡tr✐❝ ❢♦r♠✳ ◆❛♠❡❧②✱ ♣✉tt✐♥❣ t❤❡ ❝♦♠♠♦♥ ✈❛❧✉❡ ♦❢ t❤❡t❤r❡❡ r❛t✐♦s ♦❢ t❤❡ ❝❛♥♦♥✐❝❛❧ ❡q✉❛t✐♦♥ ❡q✉❛❧ t♦ t✱ ✇❡ ❣❡t

x = kt+ x0, y = lt+ y0, z = mt+ z0

✇❤✐❝❤ ❛r❡ t❤❡ ♣❛r❛♠❡tr✐❝ ❡q✉❛t✐♦♥s ♦❢ ❛ str❛✐❣❤t ❧✐♥❡✳▲❡t ✉s ✜♥❞ ♦✉t ✇❤❛t ❛r❡ t❤❡ ♣❡❝✉❧✐❛r✐t✐❡s ♦❢ t❤❡ ♣♦s✐t✐♦♥ ♦❢ ❛ str❛✐❣❤t ❧✐♥❡

r❡❧❛t✐✈❡ t♦ t❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ✐❢ s♦♠❡ ♦❢ t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ ❝❛♥♦♥✐❝❛❧❡q✉❛t✐♦♥ ❛r❡ ❡q✉❛❧ t♦ ③❡r♦✳

❙✐♥❝❡ t❤❡ ✈❡❝t♦r e(k, l,m) ✐s ♣❛r❛❧❧❡❧ t♦ t❤❡ str❛✐❣❤t ❧✐♥❡✱ ✇✐t❤ m = 0 t❤❡❧✐♥❡ ✐s ♣❛r❛❧❧❡❧ t♦ t❤❡ xy✲♣❧❛♥❡ (eex = 0)✱ ✇✐t❤ l = 0 t❤❡ ❧✐♥❡ ✐s ♣❛r❛❧❧❡❧ t♦t❤❡ xz✲♣❧❛♥❡✱ ❛♥❞ ✇✐t❤ k = 0 ✐t ✐s ♣❛r❛❧❧❡❧ t♦ t❤❡ yz✲♣❧❛♥❡✳

■❢ k = 0 ❛♥❞ l = 0✱ t❤❡♥ t❤❡ str❛✐❣❤t ❧✐♥❡ ✐s ♣❛r❛❧❧❡❧ t♦ t❤❡ z✲❛①✐s ✭e ✐s♣❛r❛❧❧❡❧ t♦ ez❀ ✐❢ l = 0 ❛♥❞ m = 0✱ t❤❡♥ ✐t ✐s ♣❛r❛❧❧❡❧ t♦ t❤❡ x✲❛①✐s✱ ❛♥❞ ✐❢k = 0 ❛♥❞ m = 0✱ t❤❡♥ t❤❡ ❧✐♥❡ ✐s ♣❛r❛❧❧❡❧ t♦ t❤❡ y✲❛①✐s✳

❲❡ ❝♦♥❝❧✉❞❡ ✇✐t❤ ❛ ♥♦t❡ t❤❛t ❛ str❛✐❣❤t ❧✐♥❡ ♠❛② ❜❡ s♣❡❝✐✜❡❞ ❜② t❤❡❡q✉❛t✐♦♥s ♦❢ t❤❡ ❢♦r♠ (∗) ❛♥❞ (∗∗) ✐♥ ❈❛rt❡s✐❛♥ ❝♦♦r❞✐♥❛t❡s ✐♥ ❣❡♥❡r❛❧ ✭❛♥❞♥♦t ♦♥❧② ✐♥ ✐ts ♣❛rt✐❝✉❧❛r ❝❛s❡✱ ✐✳❡✳ ✐♥ r❡❝t❛♥❣✉❧❛r ❈❛rt❡s✐❛♥ ❝♦♦r❞✐♥❛t❡s✮✳

✶✾✳✾ ❊①❡r❝✐s❡s

✶✳ ❯♥❞❡r ✇❤❛t ❝♦♥❞✐t✐♦♥ ❞♦❡s ❛ str❛✐❣❤t ❧✐♥❡ r❡♣r❡s❡♥t❡❞ ❜② t❤❡ ❡q✉❛t✐♦♥✐♥ ❝❛♥♦♥✐❝❛❧ ❢♦r♠ (∗∗) ✐♥t❡rs❡❝t t❤❡ x✲❛①✐s ✭y✲❛①✐s✱ z✲❛①✐s✮❄ ❯♥❞❡r ✇❤❛t❝♦♥❞✐t✐♦♥ ✐s ✐t ♣❛r❛❧❧❡❧ t♦ t❤❡ ♣❧❛♥❡ xy(yz, zx)❄

✷✳ ❙❤♦✇ t❤❛t t❤❡ ❧♦❝✉s ♦❢ ♣♦✐♥ts ❡q✉✐❞✐st❛♥t ❢r♦♠ t❤r❡❡ ♣❛✐r✇✐s❡ ♥♦♥✲♣❛r❛❧❧❡❧ ♣❧❛♥❡s ✐s ❛ str❛✐❣❤t ❧✐♥❡✳

✸✳ ❙❤♦✇ t❤❛t t❤❡ ❧♦❝✉s ♦❢ ♣♦✐♥ts ❡q✉✐❞✐st❛♥t ❢r♦♠ t❤❡ ✈❡rt✐❝❡s ♦❢ ❛ tr✐❛♥❣❧❡✐s ❛ str❛✐❣❤t ❧✐♥❡✳ ❋♦r♠ ✐ts ❡q✉❛t✐♦♥s ❣✐✈❡♥ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ✈❡rt✐❝❡s ♦❢t❤❡ tr✐❛♥❣❧❡✳

✹✳ ❙❤♦✇ t❤❛t t❤r♦✉❣❤ ❡❛❝❤ ♣♦✐♥t ♦❢ t❤❡ s✉r❢❛❝❡

z = axy

t❤❡r❡ ♣❛ss t✇♦ str❛✐❣❤t ❧✐♥❡s ❡♥t✐r❡❧② ❧②✐♥❣ ♦♥ t❤❡ s✉r❢❛❝❡✳✺✳ ■❢ t❤❡ str❛✐❣❤t ❧✐♥❡s s♣❡❝✐✜❡❞ ❜② t❤❡ ❡q✉❛t✐♦♥s

a1x+ b1y + c1z + d1 = 0,

a2x+ b2y + c2z + d2 = 0.

}

✶✾✳✶✵✳ ❇❆❙■❈ P❘❖❇▲❊▼❙ ❖❋ ❙❚❘❆■●❍❚ ▲■◆❊❙ ❆◆❉ P▲❆◆❊❙ ✷✷✺

❛♥❞a3x+ b3y + c3z + d3 = 0,

a4x+ b4y + c4z + d4 = 0.

}

✐♥t❡rs❡❝t✱ t❤❡♥∣

∣

∣

∣

∣

∣

∣

∣

a1 b1 c1 d1a2 b2 c2 d2a2 b2 c2 d2a4 b4 c4 d4

∣

∣

∣

∣

∣

∣

∣

∣

= 0.

❙❤♦✇ t❤✐s✳

✶✾✳✶✵ ❇❛s✐❝ ♣r♦❜❧❡♠s ♦❢ str❛✐❣❤t ❧✐♥❡s ❛♥❞ ♣❧❛♥❡s

❋♦r♠ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛♥ ❛r❜✐tr❛r② ♣❧❛♥❡ ♣❛ss✐♥❣ t❤r♦✉❣❤ t❤❡ ♣♦✐♥t (x0, y0, z0)✳❆♥② ♣❧❛♥❡ ✐s s♣❡❝✐✜❡❞ ❜② ❛♥ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❢♦r♠

ax+ by + cz + d = 0.

❙✐♥❝❡ t❤❡ ♣♦✐♥t (x0, y0, z0) ❜❡❧♦♥❣s t♦ t❤❡ ♣❧❛♥❡✱ t❤❡♥

a0x+ b0y + c0z + d0 = 0.

❍❡♥❝❡ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ r❡q✉✐r❡❞ ♣❧❛♥❡ ✐s

ax+ by + cz − (a0x+ b0y + c0z) = 0,

♦ra(x− x0) + b(y − y0) + c(z − z0) = 0.

❖❜✈✐♦✉s❧②✱ ❢♦r ❛♥② a✱ b✱ c t❤✐s ❡q✉❛t✐♦♥ ✐s s❛t✐s✜❡❞ ❜② t❤❡ ♣♦✐♥t (x0, y0, z0)✳❋♦r♠ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛♥ ❛r❜✐tr❛r② str❛✐❣❤t ❧✐♥❡ ♣❛ss✐♥❣ t❤r♦✉❣❤ t❤❡ ♣♦✐♥t

(x0, y0, z0)✳❚❤❡ r❡q✉✐r❡❞ ❡q✉❛t✐♦♥ ✐s

x− x0

k=

y − y0l

=z − z0m

.

■♥❞❡❡❞✱ t❤✐s ❡q✉❛t✐♦♥ s♣❡❝✐✜❡s ❛ str❛✐❣❤t ❧✐♥❡ ♣❛ss✐♥❣ t❤r♦✉❣❤ t❤❡ ♣♦✐♥t(x0, y0, z0) ✇❤♦s❡ ❝♦♦r❞✐♥❛t❡s ♦❜✈✐♦✉s❧② s❛t✐s❢② t❤❡ ❡q✉❛t✐♦♥✳ ❚❛❦✐♥❣ ❛r❜✐tr❛r②✭♥♦t ❛❧❧ ❡q✉❛❧ t♦ ③❡r♦✮ ✈❛❧✉❡s ❢♦r k✱ l✱ m✱ ✇❡ ♦❜t❛✐♥ ❛ str❛✐❣❤t ❧✐♥❡ ♦❢ ❛♥❛r❜✐tr❛r② ❞✐r❡❝t✐♦♥✳

❋♦r♠ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ str❛✐❣❤t ❧✐♥❡ ♣❛ss✐♥❣ t❤r♦✉❣❤ t✇♦ ❣✐✈❡♥ ♣♦✐♥ts(x′, y′, z′) ❛♥❞ (x′′, y′′, z′′)✳

✷✷✻ ❈❍❆P❚❊❘ ✶✾✳ ❆ P▲❆◆❊ ❆◆❉ ❆ ❙❚❘❆■●❍❚ ▲■◆❊

❚❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ str❛✐❣❤t ❧✐♥❡ ♠❛② ❜❡ ✇r✐tt❡♥ ✐♥ t❤❡ ❢♦r♠

x− x′

k=

y − y′

l=

z − z′

m.

❙✐♥❝❡ t❤❡ s❡❝♦♥❞ ♣♦✐♥ts ❧✐❡s ♦♥ t❤❡ ❧✐♥❡✱ t❤❡♥

x′′ − x′

k=

y′′ − y′

l=

z′′ − z′

m.

❚❤✐s ❛❧❧♦✇s ✉s t♦ ❡❧✐♠✐♥❛t❡ k✱ l✱ m✱ ❛♥❞ ✇❡ ❣❡t t❤❡ ❡q✉❛t✐♦♥

x− x′

x′′ − x′ =y − y′

y′′ − y′=

z − z′

z′′ − z′.

❋♦r♠ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ♣❧❛♥❡ ♣❛ss✐♥❣ t❤r♦✉❣❤ t❤r❡❡ ♣♦✐♥ts A′(x′, y′, z′)✱A′′(x′′, y′′, z′′)✱ A′′′(x′′′, y′′′, z′′′)✱ ♥♦t ❧②✐♥❣ ♦♥ ❛ str❛✐❣❤t ❧✐♥❡✳

▲❡t A(x, y, z) ❜❡ ❛♥ ❛r❜✐tr❛r② ♣♦✐♥t ❜❡❧♦♥❣✐♥❣ t♦ t❤❡ r❡q✉✐r❡❞ ♣❧❛♥❡✳ ❚❤❡t❤r❡❡ ✈❡❝t♦rs −−→

A′A,−−−→A′A′′,

−−−→A′A′′′

❧✐❡ ✐♥ ♦♥❡ ♣❧❛♥❡✳ ❈♦♥s❡q✉❡♥t❧②✱

(−−→A′A,

−−−→A′A′′,

−−−→A′A′′′) = 0,

❛♥❞ ✇❡ ❣❡t t❤❡ r❡q✉✐r❡❞ ❡q✉❛t✐♦♥∣

∣

∣

∣

∣

∣

x− x′ y − y′ z − z′

x′′ − x′ y′′ − y′ z′′ − z′

x′′′ − x′ y′′′ − y′ z′′′ − z′

∣

∣

∣

∣

∣

∣

= 0.

❋♦r♠ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ♣❧❛♥❡ ♣❛ss✐♥❣ t❤r♦✉❣❤ ❛ ❣✐✈❡♥ ♣♦✐♥t (x0, y0, z0) ❛♥❞♣❛r❛❧❧❡❧ t♦ t❤❡ ♣❧❛♥❡

ax+ by + cz + d = 0.

❚❤❡ r❡q✉✐r❡❞ ❡q✉❛t✐♦♥ ✐s

a(x− x0) + b(y − y0) + c(z − z0) = 0.

■♥❞❡❡❞✱ t❤✐s ♣❧❛♥❡ ♣❛ss❡s t❤r♦✉❣❤ t❤❡ ❣✐✈❡♥ ♣♦✐♥t ❛♥❞ ✐s ♣❛r❛❧❧❡❧ t♦ t❤❡ ❣✐✈❡♥♣❧❛♥❡✳

❋♦r♠ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ str❛✐❣❤t ❧✐♥❡ ♣❛ss✐♥❣ t❤r♦✉❣❤ ❛ ❣✐✈❡♥ ♣♦✐♥t (x0, y0, z0)♣❛r❛❧❧❡❧ t♦ ❛ ❣✐✈❡♥ str❛✐❣❤t ❧✐♥❡

x− x′

k=

y − y′

l=

z − z′

m.

✶✾✳✶✵✳ ❇❆❙■❈ P❘❖❇▲❊▼❙ ❖❋ ❙❚❘❆■●❍❚ ▲■◆❊❙ ❆◆❉ P▲❆◆❊❙ ✷✷✼

❚❤❡ r❡q✉✐r❡❞ ❡q✉❛t✐♦♥ ✐s

x− x0

k=

y − y0l

=z − z0m

.

❆ str❛✐❣❤t ❧✐♥❡ ♣❛ss✐♥❣ t❤r♦✉❣❤ ❛ ♣♦✐♥t (x0, y0, z0) ❛♥❞ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ ❛♣❧❛♥❡

ax+ by + cz + d = 0,

✐s s♣❡❝✐✜❡❞ ❜② t❤❡ ❡q✉❛t✐♦♥

x− x0

a=

y − y0l

=z − z0

c.

❆ ♣❧❛♥❡ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ ❛ str❛✐❣❤t ❧✐♥❡

x− x′

k=

y − y′

l=

z − z′

m,

♣❛ss✐♥❣ t❤r♦✉❣❤ ❛ ♣♦✐♥t (x0, y0, z0)✱ ✐s s♣❡❝✐✜❡❞ ❜② t❤❡ ❡q✉❛t✐♦♥

k(x− x0) + l(y − y0) +m(z − z0) = 0.

❧❡t ✉s ❢♦r♠ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ♣❧❛♥❡ ♣❛ss✐♥❣ t❤r♦✉❣❤ ❛ ♣♦✐♥t (x0, y0, z0) ❛♥❞♣❛r❛❧❧❡❧ t♦ t❤❡ str❛✐❣❤t ❧✐♥❡s

x− x′

k′ =y − y′

l′=

z − z′

m′ ,

x− x′′

k′′ =y − y′′

l′′=

z − z′′

m′′ .

❙✐♥❝❡ t❤❡ ✈❡❝t♦r (k′, l′,m′)✱ ❛♥❞ (k′′, l′′,m′′) ❛r❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ ♣❧❛♥❡✱ t❤❡✐r✈❡❝t♦r ♣r♦❞✉❝t ✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ♣❧❛♥❡✳ ❍❡♥❝❡ t❤❡ ❡q✉❛t✐♦♥ ✐s

(x− x0)

∣

∣

∣

∣

l′ m′

l′′ m′′

∣

∣

∣

∣

+ (y − y0)

∣

∣

∣

∣

m′ k′

m′′ k′′

∣

∣

∣

∣

+ (z − z0)

∣

∣

∣

∣

k′ l′

k′′ l′′

∣

∣

∣

∣

= 0,

✇❤✐❝❤ ❝❛♥ ❜❡ r❡✇r✐tt❡♥ ✐♥ ❛ ❝♦♠♣❛❝t ❢♦r♠✿∣

∣

∣

∣

∣

∣

x− x0 y − y0 z − z0k′ l′ m′

k′′ l′′ m′′

∣

∣

∣

∣

∣

∣

= 0.

✷✷✽ ❈❍❆P❚❊❘ ✶✾✳ ❆ P▲❆◆❊ ❆◆❉ ❆ ❙❚❘❆■●❍❚ ▲■◆❊

❈❤❛♣t❡r ✷✵

❆❝❦♥♦✇❧❡❞❣❡♠❡♥t

❙✉♣♣♦rt❡❞ ❜② ❚➪▼❖P✲✹✳✶✳✷✳❆✴✶✲✶✶✴✶✲✷✵✶✶✲✵✵✾✽

✷✷✾

✷✸✵ ❈❍❆P❚❊❘ ✷✵✳ ❆❈❑◆❖❲▲❊❉●❊▼❊◆❚

❇✐❜❧✐♦❣r❛♣❤②

❬✶❪ ❏✳ ❘♦❡✱ ❊❧❡♠❡♥t❛r② ●❡♦♠❡tr②✱ ❖①❢♦r❞ ❯♥✐✈❡rs✐t② Pr❡ss✱ ✶✾✾✸✳

❬✷❪ ❆✳ ❘é♥②✐✱ ❆rs ▼❛t❡♠❛t✐❝❛✱ ❚②♣♦t❡① Pr❡ss✱ ✷✵✵✺✳

❬✸❪ ▲✳ ▼❧♦❞✐♥♦✇✱ ❊✉❝❧✐❞✬s ❲✐♥❞♦✇✱ ❚❤❡ ❙t♦r② ♦❢ ●❡♦♠❡tr② ❢r♦♠ P❛r❛❧❧❡❧❧✐♥❡s t♦ ❍②♣❡rs♣❛❝❡✱ ❚♦✉❝❤st♦♥❡✱ ◆❡✇ ❨♦r❦✱ ✷✵✵✷✳

❬✹❪ ●✳ Pó❧②❛✱ ▼❛t❤❡♠❛t✐❝❛❧ ▼❡t❤♦❞s ✐♥ ❙❝✐❡♥❝❡✱ ❚❤❡ ▼❛t❤❡♠❛t✐❝❛❧ ❆ss♦❝✐✲❛t✐♦♥ ♦❢ ❆♠❡r✐❝❛✱ ❲❛s❤✐♥❣t♦♥✱ ✶✾✼✼✳

❬✺❪ ❍✳ ❙✳ ▼✳ ❈♦①❡t❡r✱ ■♥tr♦❞✉❝t✐♦♥ t♦ ●❡♦♠❡tr②✱ ❏♦❤♥ ❲✐❧❡② ❛♥❞ ❙♦♥s✱ ■♥❝✳✱❙❡❝♦♥❞ ❊❞✐t✐♦♥✱ ✶✾✻✾✳

❬✻❪ ❍✳ ❆♥t♦♥✱ ❈❛❧❝✉❧✉s ✇✐t❤ ❛♥❛❧②t✐❝ ❣❡♦♠❡tr② ✴ ▲❛t❡ ❚r✐❣♦♥♦♠❡tr② ✈❡rs✐♦♥✱❚❤✐r❞ ❡❞✐t✐♦♥✱ ❏♦❤♥ ❲✐❧❡② ❛♥❞ ❙♦♥✱ ✶✾✽✾✳

✷✸✶