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