In the figure below, given a triangle
ABC, construct the incircle with incenter I and the excircle with
excenter E. Let be D and F the tangent
points of triangle ABC with its excircle. ID and BC meet at G,
and IF and BC meet at H. If S1, S2, and S3 are the areas of the triangles
BDG, CFH, and
respectively, prove that S1 + S2 = S3.
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