In the figure below, given a triangle
ABC, construct the incenter I and the excircles. Let be D,
E, F, G, H, and J the tangent
points of triangle ABC with its excircles. K, M, N, P, Q, and R
are the intersection points of triangle ABC and ID, IE, IF, IG,
IH, and IJ respectively. If S_{1}, S_{2}, S_{3},
S_{4}, S_{5}, S_{6}, S_{7}, S_{8},
and S_{9}, are the areas of the shaded triangles, prove that
S_{1}+S_{2}+S_{3}+S_{4}+S_{5}+S_{6}
= S_{7}+S_{8}+S_{9}.
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