The figure below shows a triangle ABC with the excenters E_{1}, E_{2}, and E_{3}.
Lines AE_{1}, BE_{2}, and CE_{3} meet
circumcircle O at A_{1}, B_{1}, and C_{1},
respectively. If S_{1}, S_{2}, and
S_{3} are the areas of triangle E_{1}E_{2}E_{3},
hexagon AC_{1}BA_{1}CC_{1},
and triangle A_{1}B_{1}C_{1},
respectively, prove that
S_{1} = 2.S_{2} = 4.S_{3}.
See also:
Kaleidoscope of Problem 1112 base on Poincare Disk Model.
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