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Problem 567: Right triangle, Incircle, Excircle, Collinear tangency points.
Level: High School, College, Math Education.

The drawing shows a right triangle ABC. The incircle D is tangent to AB and AC at G and H, respectively. The excircle E is tangent to CB extended at F. Prove that F, G, and H are collinear points.

Right triangle, Incircle, Excircle, Collinear tangency points

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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George Pólya's 1945 book "How to Solve It, A new aspect of Mathematical Method", is a book describing methods of problem solving. It suggests the following steps when solving a mathematical problem: (1) First, you have to understand the problem. (2) After understanding, then make a plan. (3) Carry out the plan. (4) Look back on your work. How could it be better? 

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