179 chuyãšn ä Ề toã-n tuá»”i thæ 2 - thcs
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Tinh chat sau day cUa trvc tam tam giae co nhieu Lmg dl,lng trong vi~e giai cac bal tap hlnh hQC.
Tinh chat. Cho AABC co H la tnrc tam, 0 la tam dlti1ng tron nqoal tiep va M la trung diem BC. The thl AH = 20M.
Chung minh. Xet tntong hop AABC nhon (cac tntong hop khac ehLmg minh tuang tl/).
Va duong kinh BD ella (0),
VI DAB = 90 nan DA.l AB. Ma CH .1 AB nan DA II CH. Tuang tl,l DC II AH. Suy ra t(( giae DAHC la hlnh blnh hanh. Do d6 DC =AH. Ma DC =20M (vl OM la duong trung binh ella
aDBC) nan AH = 20M (dpcm),
Bili t~p ap dl,lng Bili toan 1. Cho AABC nhon ni tiep duong
tron tam O. G9i M, N va Plan luqt la trung diem cac canh BC, CA va AB. Chltng minh rang cac duong th~ng qua M song song vai OA, qua N song song vai OB va qua P song song vai OC dong quy. ._
Loi giai. G9i H la true tam AABC; I va Elan luqt la trung diem ella HA va HO.
ViAH = 20M nan AI = IH = OM. Ma AH 110M (do eung vuong g6e vai BC) nan
cac t(( giae AIMO va IHMO la hinh blnh hanh.
.
T~ MOT TINH CHAT VE TR~C TAM TAM 61Ac
D';NG HAl GIANG
(GV THCS Thl tran C~m Xuym, HiJ Tinh)
Suy ra MI II AO va E la trung diem ella MI. T((e la E thuoc duong th~ng qua M song song
vai OA. Tuang tl,l E cling thuoc cac duong th~ng qua I\J
song song vai OB va qua P song song vai OC. Tlt d6 suy ra dpem.
Bid toan 2. Cho AABC nhon e6 H la tnrc tam. Chltng minh rang 9 diem g6m chan ba duong cao, trung diem ba canh va trung diem cac doan HA, HB, HC dong viano
LOi giai. sadunq hinh ve va ehltng minh ella biJi teen 1 ta co:
1 1ED =EM =EI = -MI = -OA.
2 2 Sa dl,lng cac ket qua tuang tl,l va ket hop vai
OA = OB = OC ta suy ra 9 diem da eho cling thuoc duong tron tam E.
Bili toan 3. Cho AABC nhon ni tiep duong tron tam 0 co cac duong caoAN va CK. Duong tron nqoal tiep aBKN cat duong tron (0) tal diem th(( hai M. G9i I la trung diem ella AC. Chltng minh 1M .1 MB.
LOi giai. G9i H la tnrc, tam ella AABC, J la trung diem BH.
Ta thay J la tam duong tron ngo~i tiep aBKN va BH la duong kinh ella (J).
t
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VI IHJO la hlnh blnh hanh nan IH II OJ. M~t khac, VI M, B la giao di~m ella hai duong
tron (0), (J) nan OJ 1- MB. Suy ra IH 1- MB. Ma BH la duong kinh ella (J) nan HM 1- MB. Tli d6 suy ra \, H, M th~ng hang va 1M 1- MB. Bcii toan 4. Cho tCt giae ABCD ni tiep duong
tron (0 ; R). GQi H1, H2, H3, H4 thCt nr la tnrc tam
ella cac tam giae ACD, BCD, ABD va ABC. ChUng minh rang: a) BH1,AH2, CH3 va DH4 dong quy.
b) Bon di~m H1, H2, H3, H4 dong viano
H3 Or H.
D
Li:fi giai. a) Vi AH1= BH2 (= 20M) va AH1II BH2 (vl cling vuong g6e vai CD) nan tCt gia~ AH1H2B la hlnh blnh hanh,
Suy ra AH2 va BH1 cat nhau tai trung di~m I
ella moi duong. ChUng minh tuang nr vai cac cap di~m khac ta
suy ra BH1,AH2, CH3 va DH4 dong quy tal trung
di~m I ella moi duong (dpem).
Vi tCtgiaeDOH4011ahinh blnh hanh nan 01H4 :
= OD = R. : ChUng minh tuang nr ta c6 :
01 H1 =01 H2 =01H3 =R. : V~y bOn di~m H3, H4 cilng thuQc : H1, H2,
dLtOng tron tam 0 1 ban kinh R (dpcm).
Bid toan 5. Cho dLtOng tron tam 0 ban kinh R : va d~m P c6dinh narn ngoai dl.tOng tron. Va M'p : tuyen PAva cat tuyen PBC (A, B, C nam trEm (0)). : Chang minh rang khi cat tuyen PBC thay d6i thi :trve tam H ella MBC chay trim mt dLtOng cO dinh. : M :
Li:fi gild. GQi I la hlnh ehieu ella 0 tren BC, K la trung di~m ella AH. Lay 0' doi xUng vai 0 qua trung di~m ella PA.
Ta thay 0' la di~m cO djnh va tCt giae AOPO' la hlnh blnh hanh, Suy ra O'P II AO va O'P = AO. COng vi tlt giae AOIK la hlnh blnh hanh nan AO II KI va AO = KI. Suy ra O'P II KI va O'P = KI. Tli d6 tlt giae O'PIK la hlnh blnh hanh, Suy ra O'K II PI. Ma PI 1-AK nan O'K 1-AK. Suy ra ~O'AH can tal 0' ~ O'H = O'A. Tli d6 H thue duong tron cO djnh tam 0' ban
kfnh O'A (dpem). Bcii ~p til luy~n. Bcii 1. ChUng minh rang tnrc tam, tronq tam va
tam duong tron nqoal tiep ella mt tam giae cling . nam tren mt duong th~ng (duong th~ng Ole).
Bcii 2. Cho duong tron (0) va ba di~m A, B, C thay 95i tren (0) sao eho tnrc tam H ella MBC la mt di~m co dinh nam trong (0). Tim quy tieh b) Lay 0 1 doi xUng vai 0 qua I. chan cac duong cao ella MBC. .....................................................................
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" Tim otto ut 00 ilieu bien
THAI HOU HU~ (Gv. THCS Quang U:,c, Can Lt?c, Ha Tfnh)
Co rat nhieu phLlong phap giai mt?t bet toen bat dlmg thLic (BDT), nhLlng chUng ta can tim mt?tphLlong phap giai ng~n 99n nhat. D~c bi~t, chUng ta can t
-
.:xL ~ Co ehinh phttong hhong? Cho so t'! nhien S c6 2011 chlt so, trong d6 c6 2010 chlt so 5 va mt chlt so
a khac 5. Hei S c6 phai la so chfnh phuong hay khOng?
CAO QUaC CUONG (GV THCS Vinh Tuong, Vinh Pnuc)
------------------------~---~,.- .... ""7/ ~T... ? [L'IO -/~ t: I\lRO ~ " --A.EL qua _"
Cach dt,tng. Dlfng mot duong tron tam . cat
Ox, Oy tuong ltng tal cac diem A1' A2.
Tren (0) dlfng lien tiep cac diem cac diem A3,
A4 , , A20 phan bi$t, khac A1,A2 thea man A2A3 = A3A4 =... = A 19A20 = A1A2
Tren cung A1A2 cua (0) d,!ng lien tiep cac
diem cac diern B1, B2,., B17 phan biet, khacA1, A20 thea man
B2B3 =... = B16B17 =A1A20. A20B1 = B1B2 =
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