Higher Euclidean Geometry Lachlan 1893

Transcript of Higher Euclidean Geometry Lachlan 1893

http://slidepdf.com/reader/full/higher-euclidean-geometry-lachlan-1893 1/310
a
triangle
112
as the
always turning
but
it
may
be
assumes
when
P'
is
made
to
approach
indefinitely
near
to
P
points
present
treatise
we
problem we should
cut
point
0,
and
may
be
same
drawn
measured. It
Is usual
may
be
B,
AO
be
produced
to
A';
then
sin
AOB
the
point
a towards
22.
\BA.BC.amABC;
or
by
triangle,
but
when
the
OX, OA,
straight
line,
AB
has
the
pencil,
and
&c.
From
A
draw
AB
equal
and
(POA)
n
points
in
a
plane,
and x any
;
point
joining the middle
point
of
divided
harmonically
in
the
points
P
and
Q.
Q
Thus,
the
the
bisectors
of
the
angle
BAC
divide
drawn to
And
P.
44.
Ex.
conjugate
of
A
with
respect
For
since
meeting
OA,
OB
in
A'
and
pencil
if P
harmonic
last.
49.
Any
a
harmonic
range,
and
the
range
{PQ',
R'tf}.
Take
straight
that
the
show
that
OA
may
be
Q
respectively
with
respect
to
A
and
B,
prove
that
A HARMONIC
which
as
2
intersect
in
0'.
By
the
last
article,
the
line
joining
the
harmonic
conjugates
of
P
with
respect
to
line, two
such
that
their
distances
segments A A', BB',
range
such
that
the
straight line, show
respect
73.
Ex.
1.
If
middle
points
in
involution
that
A,
A',
be
ranges
tan
2
XOS
rays are
the
rays
of
a
pencil
in
involution
constitute
drawn
perpendicular
to
the
the
harmonic
conjugate
are
at
right
angles,
show
that
they
bisect the angle between each pair of conjugate rays of the
pencil.
Ex.
6.
Show
rays
rays
between a
...,
Let A
and
let
pencil.
By
a
rays
0{AA',
BB',...}, in
then
by
any
pencil
in
involution
[A
A',
BB',
GG'\
0,
rays
OX,
OY
OZ
that
geometry, any
the
vertices
with
a
triangle.
In
recent
0,
and
let
THROUGH THE
pencil
of
a
the
lines
drawn
from
the circunicentre
are isogonal
points
L,
M,
N,
BAC,
is
the
line
joining
A
to
the
the
lino
DC
so
that
the
segment*
XX',
BC
have
the
ABC touch the
the
escribed
.
in
collinear
points.
Ex.
The
lines
BE,
CF
collinear
points.
109.
arbitrary
point, show
that the
0,
the
line
polar
by
joining
segments intercepted on them are equal.
Ex.
7.
middle
the
points
X,
X,
the
line
joining
the
median
points
of
the
triangles
ABC,
A'
B'C
touches
AB
at
B.
This
theorem
middle
points
of
the
sides,
the
lines
drawn
through
OF A
perpendiculars
from
A,
B,
C
A,
corresponds to
Q,
Q'
are
equidistant
with
respect
to
a
orthocentre
XZDF
and
XED
Y
since
XPD,
YQE,
the
ACB, are concurrent.
with
this
circle,
it
follows
at
suppose,
is
parallel
triangle
in
proportional
to
BC
a
Tucker
circle.
mentioned in
a paper
by Mr
similar
to
the
triangle
ABC.
groups
of
lines.
And
such
figures
we
shall
call
the
word
opposite.
Thus,
in
the
tetrastigm
ABCD
the
connector
CD
is
of
a polystigm
of 2»
points has
intersection
of
the
four
will
intersect
on
BD.
Ex.
6.
The
connectors
ABCD meet in
G a
to
the
triangle
PQR
intersect
the
corresponding
PL},
§
show
points, so
of
a
tetragram,
show
that
AC.
AC
the
tetragram, and
if the
line joining
any
transversal
cuts
the
lines
a,
b,
c,
d,
will
; B,
B'
of
Y
and
Z
with
respect
to
line
joining
points
which
lie
CHAPTER VIII.
the
will
intersect
in
a
line
(L
say).
Also
lie
A'B'C.
See
passes
through
G.
Ex.
7.
Two
sides
are concurrent
in
perspective
of
per
spective.
o--»----_v.
Let
be
the
centre
of
perspective
of
the
they
Z
being
collinear
points.
of
the
J>,
h,/b.
Therefore
lines, three
known as
it
that
ZX,
have
the
X YY', X'Y Z
point
which
is
also
with
the
triangle
formed
by
the
are
taken
the
points
X,
of
connectors
of
a
hexastigm
inscribed
in
F
be
any
six
points
on
a
circle.
Lot
AD,
triangles have the same
(AB\ (CE\ (DF\
through
sixty
points.
This
theorem
is
known
will
be
concurrent
/A
TiC\
(
~
,_,„]
of
figure
F,
coincides
projection
F'
on
some
plane,
being
the
vertex
of
projection.
Let
us
take
in
;
a
figure
CHAPTER IX.
THE THEORY
CAB are respectively
equal to the
angles C'B'A', A'C'B',
B'A'C, the triangles
Hence,
it
appears
that
to
the
triangle
ABC.
Thus
is
the
centre
of
similitude
of
the
be
drawn
parallel
to
any
of
the
figure
F'.
This
also
follows
homothetic
to
0,
corresponding
to
a.
XOX'.
216.
Directly
INVERSELY SIMILAR.
points
of
a
figure
F,
and
A',
R,G'
are
inversely
of concurrence, show
ABC,
Then
we
have
Hence
I
u
I
9t
I
3
2
x
3
and
the
triangle
of
similitude,
show
that
the
line
P
X
P
2
P
3
P
x
lies
triangle
CB',
AC
are
corresponding
lines,
and
Show
If
G
be
with
the
triangle
A'
B'C.
Ex.
2.
Show
that
the
circumcentre
A , B ,
B'C
in
points
always
at
a
constant
distance
from
Let
class,
The
envelope
of
a
straight
line
which
moves
in
and magnitude
POLES AND
Let the tangent
Therefore
way
that
{T
V,
CD}
chord
points.
Further,
tangent
respect
to
the
these
lines
triangle
ABC,
show
that
PB
is drawn through the pole of the line BC, with
respect
Q
on
the
tangent
at
the point
C
and
D'
: show
that
CD
passes
CONJUGATE TRIANGLES.
orthogonally.
Conjugate
triangles.
262.
The
triangle
formed
by
A,
B,
C,
respectively,
A
in
the
polar
circles
of
the
four
triangles
ABC,
BOC,
CO
centre of
diameter,
show
that
272.
Let
A
BCD
be
follows that
B
P';
the
any circle
circle
form
a
self-conjugate
triangle
line joining
have
been
three centres
of
a
tetrastigm.
of a
in
the
points
X,
X'
intersection
of
have
points
with
respect
to
the
the
the
pencil
in
of
we
have
the
theorem
AND BRIANCHON'S
of
CHAPTER XL
F,
we
touch,
the
corresponding
perspective,
their
be
(§
of
178 RECIPROCATION
Oa.OA
line
we
have
tangents
will
have
the
theorem :
If
a
chord
of
a
conic
be
the
line
subtends
conjugate
line
joining
the
centres
of
two
circles,
such
that
its
to
that
the
locus
of
the
186 CONSTRUCTION
ABC, show
radical
axis
of
the
triangle and
the Lemoine
circle of
symmedian
real points.
L
and
L'
vertices
of
a
tetragram,
AB.
Then
two given circles
and a
the
circle
the
the side
BG in
CENTRES OF
AX';
BY,
BY';
the
of the
Q.
pairs
of
circles
A,
X
the
given
circles
in
R.
Let
touching
described with
for centre,
PROPERTIES OF
of
polar
circles
of
the
by
the
the
three
pairs
of
circles,
will
be
coaxal.
Ex.
3.
Show
that
the
square
of
circle
will
cut
orthogonally
a
circle
of
if P, P' be the points of contact of .5(7
with
the
two
circles
of
the
system
which
Ex.
9.
The
sides
of
the
triangle
ABC
locus
perpendicular
subtend
a
right
angle
show that
INSCRIBED IX
that
L
must
be
one
of
these
limiting
B'
coincides
with
either
the
system.
Let
A
BCD
be
q,
C lie
inscribed
in
a
circle
X,
and
a
a fixed
circle
X.
X,
and
X
2
must
have
its
centre
PORISTIC SYSTEM
OF COAXAL
the
vertices
of
ABC,
we
see
lt
X
3
or
all
four other
circle
nectors
which
intersect
in
a
limiting
respect
peaucellier's cell.
really
taken
on
the
inverse
point
with
respect
to
a
circle
whose
centre
is
0.
Let
0A
be
the
perpendicular
from
circle which
orthogonally.
Ex.
6.
Show
that
McCay's
circles
point
P
§
circle and its
the
circle.
cut
them
in
the
PQR,
where
P,
Q,
R
are
the
points
in
which
the
lines
with
sign.
It
is
easy
to
see
that
the
ratio
(SX)
between and
inverse
circles
X,
to S,
b, a',
;
COAXAL CIRCLES.
the
corresponding
that
the
converse
theorem,
and
shall
show
that
the
converse
theorem
is
method
of
inversion
to
a
system
of
coaxal
simple
figure.
372.
Ex.
1.
Every
circle
which
touches
two
given
straight
be
inverted
;
AG,
BG,
CG
p.
73.]
Ex.
3.
Three
circles
are
drawn
through
any
the
radical
centre
ortho-
gonally.
circle
bisected by
each of
vertices
inversion
is
imaginary.
Hence,
if
we
adopt
the
above
rule
of
sign
as
a
convention,
we
may
say
that
the
inverse
circles
X,
Y,
Z
called a
and Z at the given angles, these circles will evidently
be the
touch
one
of
the
must
pass
Y,
Z.
Again,
let
the
tangents
T'
the
pole
of
this
to
the
radical
centre
of
of the
(ii)
last
article
subsist
§
system
X
lt
X
2
circles
Y,
Y
Hence,
and
let
circles
X
2
CIRCULAR TRIANGLE.
A'BC,
A'
EC,
are
given
by
the
scheme
EC,
properties,
let
01,
(01);
12,
(12);
common
tangent
circle
which
touches
the
former
in
internally
and
external
circle
of
reciprocation
is
an
imaginary
P
circles
with
respect
ratio is
when
the
circle
of
inversion
is
imaginary,
the
same
sign.
Assuming
the
convention
in the
and
negative
circles
of
whose
points
R,
point
parallel
to
MX,
the
polar
of
0,
cutting
the
lines
NP,
NP'
in
F
between N
Q'
cut
X,
X',
whose
diameter
is
OG,
where
C
is
the
centre
to this circle
G'
to
Q'.
413.
Let
X,
the pair reciprocal
F
touch
the
circle the
reciprocal circles
§
system
of
:
CD'} are
AX}=0
G
A'B
in
PROPERTY OF
any point on
Ex. 9.
Four fixed
whose
pairs
of
AC'}.
Ex. 4. On the tangent at
any point
such
ranges will
pencil
inscribe
X,
Y,
Z
be
the
homothetic
centres
of
the
given
circles
on
this
the
Trans.
Royal
Irish
be
ue
an
elegaut
proof
of
a
tetragram

Higher Euclidean Geometry Lachlan 1893

Documents

Transcript of Higher Euclidean Geometry Lachlan 1893

Lachlan Mv

Belyaev, O.a. - Fundamentals of Geometry Euclidean, Euclidean) - 2005, 231s

EUCLIDEAN DISTANCE GEOMETRY AND APPLICATIONSliberti/dgp-siam.pdf · EUCLIDEAN DISTANCE GEOMETRY AND APPLICATIONS ... Euclidean distance geometry is the study of Euclidean geometry

Euclidean and Non-Euclidean Geometry

Euclidean and Non Euclidean

Isssuuuuuuuuuuuuuuuuu lachlan y boby

Euclidean Reality

Math 333 – Euclidean and Non-Euclidean Geometry Dr. Hamblin.

MATH 3210: Euclidean and Non-Euclidean Geometryszendrei/Geom_S20/lec-04-06.pdf · Hilbert Planes with (P) and Euclidean Planes MATH 3210: Euclidean and Non-Euclidean Geometry. Propositions

7 euclidean&non euclidean geometry

RY...FORMERMEMBERS OFTHEBOARDOFTRUSTEES GeorgeE.Adams* 1893-1917 OwenF.Aldis* 1893-1898 AllisonV.Armour* 1893-1894 EdwardE.Ayer* 1893-1927 JohnC.Black* 1893-1894 M.C.Bullock* 1893-1894

Lachlan - LTS12

Non-Euclidean Geometry Topics to Accompany Euclidean and ...

MA232A: Euclidean and non-Euclidean Geometry Michaelmas ...

Robert LACHLAN (1893) AN ELEMENTARY TREATISE ON MODERN PURE GEOMETRY.pdf

Dalamerian Lachlan transition

Euclidean Geometry

Euclidean Curve Theory - library.wolfram.comlibrary.wolfram.com/infocenter/MathSource/9115/EuCurvesv3.pdf · Euclidean Curve Theory ... Alfred Gray presented Euclidean differential

Lachlan M.D. - Synowie Boga.pdf

EUCLIDEAN GEOMETRY: GRADE 12 12 Axiz...EUCLIDEAN GEOMETRY: GRADE 12 7 EUCLIDEAN GEOMETRY: GRADE 12 8 EUCLIDEAN GEOMETRY: GRADE 12 9 EUCLIDEAN GEOMETRY: GRADE 12 10 FEBRUARY - MARCH