Note. — If  0  is  ou  the  circle  ZZ'C  the  pedal  triangle  is  isosceles,
similarly  if  it  is  the  point  of  intersection  of  the  circles  ZZ'C  and
YY'B  it  is  isosceles  in  a  double  aspect,  i.e.  equilateral.
Hence  we  may  infer  that  the  circles  AXX',  BTY\  and  CZZ'  pa^s
throxtgh  two  points  0  and  0'  which  are  inverse  (Ex.  22)  ivith  respect
to  the  circum-circle  of  ABC  and  whose  pedal  triangles  toith  respect  to
ABC  are  equilateral.]


*  Le  cercle  d'Apollonius  du  triangle  ABC  par  rapport  kAB.    V.  Edtic,
Times,  Dec,  1890.
CHAPTER  VIL

POLES  AND  POLAES  WITH  RESPECT  TO  A  CIECLE.
Section  I.
Conjugate  Points,  Polar  Circle.
73.  Def.  The  perpendicular  to  the  line  joining  a  pair  of
inverse  points  passing  through  either  is  the  Polar  of  the
other  with  respect  to  the  circle.  In  the  figure  of  Art.
74  6'  and  Z  are  inverse  points ;  and  C  and  the  line
AB  are  termed  Pole  and  Polar  with  respect  to  the  circle.
Any  point  A  or  B  on  the  polar  is  the  Conjugate  of  G,
hence  the  polar  of  a  point  is  the  locus  of  its  conjugates.
Again,  since  the  circle  on  BC  as  diameter  passes  through
Z  and  therefore  cuts  the  given  one  orthogonally : —
1'.  The  circle  descHbed  on  the  line  joining  two  conjugate
Ants  cuts  the  given  circle  oi^thogonaUy .  2\  The  distance
ttween  two  conjugate  points  is  equal  to  twice  the  length
of  the  tangent  to  the  circle  from  the  middle  point  of  the
line  connecting  them.
74.  Theorem. — For  any  two  conjugate  points  B  and  C,
>  prove  that  each  lies  on  the  polar  of  the  other  ivith

respect  to  the  circle.
