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 S1, S2, S3,
and S4 are the areas of the triangles AIG, BEH, BDG, and
respectively, prove that S1 + S2 = S3
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