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 S_{1} and S_{2} 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 illstructured problem. By “construction,” we mean adding geometric figures (points, lines, planes) to a problem figure that wasn’t mentioned as "given.
