Consider triangle ABC with an inscribed circle tangent to the sides AB, BC, and AC at points D, E, and F, respectively. Let G, H, and M denote the excenters relative to sides AB, BC, and AC, respectively. Given that the areas of the triangular regions ADG, BEH, and CFM measure 10, 11, and 6 units, respectively, determine the area of triangle ABC.
Inscribed circle's charm,
Excenters lead the triangle,
Area's hidden tale.
With three regions as our guide,
Geometric puzzle unraveled.
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