In the figure below, given a triangle
ABC of area S, construct the incircle with incenter I and excircles with
excenters E1, E2, and E3. Let be D, E, F, G, H,
and M the tangent
points of triangle ABC with its excircles. If S1, S2, S3, S4, S5,
and S6 are
the areas of the triangles ICD, ICE, IAF, IAG, IBH, and IBM
respectively, prove that S1 = S2 = S3 = S4
= S5 = S6 = S/2.
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