In the figure below, given a triangle
ABC, 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, S'1, S2, S'2, S3,
and S'3 are
the areas of the shaded triangles, prove that S1 = S'1,
similarly S2 = S'2, and S3 = S'3.
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FACTS AND HINTS:
Geometry problem solving is one of the most challenging skills for students to learn. When a
problem requires auxiliary construction, the difficulty of the problem increases drastically, perhaps because deciding which construction to make is an ill-structured problem. By “construction,” we mean adding geometric figures (points, lines, planes) to a problem figure that wasn’t mentioned as "given."