Online Geometry Problem 807: Right Triangle Area, Incircle, Tangency Points, Semicircle Area, Diameter. Level: High School, SAT Prep, College, Mathematics Education

The figure below shows a right triangle ABC of area S with the incircle I tangent to AB, BC, and AC at D, E, and F, respectively. If S₁ and S₂ are the areas of semicircles of diameters AD and CE, prove that .
 

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.