Problem 85. Contact Triangles Areas, Incircle, Excircle.
Level: High School, College, SAT Prep.

In the figure below, given a triangle ABC, S₁ is the area of contact triangle of the incircle, S_a, S_b, and S_c are the areas of the contact triangles of the excircles. Prove that:
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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."