18.  Draw  two  parallel  lines  and  describe  a  number  of  circles
touching  the  lines  and  each  other  in  succession.  Invert  this
system  from  a  point  on  a  diameter  of  any  circle  perpendicular  to
the  lines  and  deduce  the  following  theorem  : —


Z  BYX


A,  B,  C  are  three  collinear  points,  and  circles  X,  T,  Z  are
described  on  the  segments  BC,  CA,  AB  respectively.  A  system  of
circles  is  drawn  as  in  figure  to  touch  each  other  and  the  given  ones,
if  C„,  p  denote  the  nih.  circle  to  prove  that  the  distance  of  its
centre  from  AB=2np.     (Pappus.)


258  INVERSION.
19.  If  three  circles  Ar^,  Br<^,  Cr^  touch  one  another  in  pairs  ;
prove  by  inversion  that  the  radii  of  the  circles  which  touch  them
with  contacts  of  similar  species  are

2rir2±2A
where  2A  is  the  area  of  the  triangle  ABC.
[Invert  from  the  point  of  contact  of  Br^^,  Cr^  with  a  radius  equal
to  the  tangent  to  Ar^;  etc.]

20.  The  rectangle  under  the  distances  of  the  ex-centre  of  simili-
tude of  two  circles  from  their  radical  axis  and  in-centre  of  simili-
tude is  equal  to  the  constant  product  of  antisimilitude.
[The  circle  of  similitude  inverts  from  either  centre  of  similitude
into  the  radical  axis  of  the  given  circles.
20a.  Prove  that  the  poles  of  the  radical  axis  of  two  circles  with
respect  to  the  circles  are  harmonic  conjugates  with  respect  to  the
centres  of  similitude.
