Geometry Problem 1317: Triangle, Excircle, Chord, Tangent, Midpoint, Arc, Sum of two Segments, Congruence.

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In the given figure, triangle ABC has an excircle O that corresponds to side BC. The points where the excircle is tangent to the sides of the triangle are labeled D, E, and F. Let M be the midpoint of the arc DE. Chords BM and CM intersect the chord DE at points G and H, respectively. You need to prove that the sum of lengths DG and HE is equal to the length GH.