In the figure below, given a triangle
ABC, construct the excircles with
excenters P and Q. Let be D and E the tangent
points of triangle ABC with its excircles. PE and DQ meet at F,
BE and DQ meet at H, PE and BD meet at G. If S_{1}, S_{2}, S_{3},
and S_{4} are the areas of the triangles BPG, EFH, DFG, and
BHQ
respectively, prove that S_{1} + S_{2} = S_{3}
+ S_{4}.
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