The figure below shows an equilateral
triangle ABC of area S. P is any point and PD, PE, and PF are perpendicular to AB, BC,
and AC, respectively. If S_{1}, S_{2}, S_{3}, S_{4}, S_{5}, and S_{6} are the areas of the shaded regions,
prove that S_{1}+S_{3}+S_{5} = S_{2}+S_{4}+S_{6} = S/2. This entry contributed by Ajit Athle.
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."
