Geometry Problem 85. Contact Triangles Areas, Incircle, Excircle, Contact Triangles. High School, College, Math Education

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: \(\dfrac{1}{S_1}=\dfrac{1}{S_a}+\dfrac{1}{S_b}+\dfrac{1}{S_c}\).
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HINTS:

1. See Problems 82, 83, 84.