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 S_{1}, S_{2}, and S_{3} are the areas of the triangles
BDG, CFH, and
IGH
respectively, prove that S_{1} + S_{2} = S_{3}.
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