﻿986 
  JS 
  T 
  ew 
  Solution 
  to 
  an 
  Historical 
  Theorem 
  in 
  Geometry. 
  

  

  From 
  Q 
  drop 
  perpendiculars 
  on 
  the 
  same 
  side 
  of 
  CO 
  on 
  to 
  

   this 
  chord 
  and 
  on 
  to 
  one 
  of 
  the 
  equal 
  chords 
  XY, 
  WZ. 
  

  

  Then 
  XY 
  (or 
  WZ) 
  > 
  or 
  < 
  MN 
  according 
  as 
  it 
  is 
  nearer 
  

   to 
  or 
  further 
  from 
  Q 
  than 
  MN, 
  a 
  fortiori 
  XY 
  (or 
  WZ) 
  > 
  or 
  

   < 
  RS. 
  

  

  Therefore, 
  through 
  any 
  point 
  in 
  the 
  bisector 
  of 
  an 
  angle, 
  

   two 
  and 
  only 
  two 
  equal 
  straight 
  lines, 
  terminated 
  by 
  the 
  arms 
  

   of 
  the 
  angle, 
  may 
  be 
  drawn 
  and 
  the 
  segments 
  into 
  which 
  

   the 
  lines 
  are 
  divided 
  at 
  the 
  point 
  are 
  equal, 
  each 
  to 
  each. 
  

  

  Q.E.D. 
  

  

  Cor. 
  If 
  the 
  internal 
  (or 
  external) 
  bisectors 
  of 
  the 
  base 
  

   angles 
  of 
  a 
  triangle 
  are 
  equal, 
  the 
  triangle 
  is 
  isosceles. 
  

  

  [Since 
  the 
  bisectors 
  of 
  the 
  angles 
  of 
  a 
  triangle 
  are 
  con- 
  

   current, 
  in 
  the 
  figure 
  the 
  bisectors 
  AD, 
  BE 
  pass 
  through 
  I 
  a 
  

  

  point 
  on 
  the 
  bisector 
  of 
  C. 
  

  

  /. 
  IA 
  = 
  IB 
  (by 
  Theorem) 
  

  

  .-. 
  IBA 
  = 
  IAB 
  

  

  .-. 
  CBA=CAB.] 
  

  

  Those 
  interested 
  in 
  the 
  theorem 
  are 
  referred 
  to 
  ' 
  Nouvelles 
  

   Annales 
  de 
  Mathematiques,' 
  1842, 
  pp. 
  138, 
  311 
  ; 
  Lond 
  v 
  Edin., 
  

   and 
  Dublin 
  Philosophical 
  Magazine, 
  1852, 
  p. 
  366 
  ; 
  1853, 
  

   April, 
  May 
  and 
  June 
  ; 
  1874, 
  p. 
  354 
  ; 
  Lady's 
  and 
  Gentleman's 
  

   Diary, 
  1857, 
  p. 
  58 
  ; 
  1859, 
  p. 
  87 
  ; 
  1860, 
  p. 
  84. 
  

  

  