
The figure below shows a circle of radius r,
chords AB and CD are parallel, chords BC and DE parallel. If
angle ABC = 45 degrees, AB = a, BC = b, CD = c, and DE = d,
prove that a^{2} + c^{2} = b^{2} + d^{2} = 4r^{2}.
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."
