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 S_{1} and S_{2} are the areas of semicircles of diameters AD
and CE, prove that
.