In the figure below, given a triangle
ABC, construct the incenter I and the excircles with
excenters P and Q. Let be D and E the tangent
points of triangle ABC with its excircles. IE and BC meet at H,
and ID and AB meet at G. If S_{1}, S_{2}, S_{3},
and S_{4} are the areas of the triangles AIG, BEH, BDG, and
CHI
respectively, prove that S_{1} + S_{2} = S_{3}
+ S_{4}.
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