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Problem 560: Triangle, Cevian, Incenter, Incircle, Perpendicular, Tangent.
Level: High School, College, Math Education

The figure shows a triangle ABC with a cevian BD. Points E, F, and O are the incenters of triangles ABD, BCD, and ABC, respectively. If EM, OG, and FH are perpendicular to AC, prove that DM = GH.

Triangle, Cevian, Incircles, Tangent, Congruence

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."

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