In the figure below, given a triangle
ABC of area S, construct the excircles with
excenters P, Q, and R. Let be D, E, F, G, H,
and M the tangent
points of triangle ABC with its excircles. If S_{1}, S_{2}, S_{3}, S_{4}, S_{5},
and S_{6} are
the areas of the triangles AMR, BFR, BEP, CHP, CGQ, and ADQ
respectively, prove that S_{1} = S_{2} = S_{3} = S_{4}
= S_{5} = S_{6} = S/2.
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