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 incircle. ED and BC meet at G,
and EF 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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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."