Here's another one:

Inscribed "Star of David" proof of the Pythagorean Theorem:

http://stuartmanderson.home.comcast.net/images/diagram.png

Let a pair of mirror congruent right triangles ABC and DEF be inscribed in a circle with diameters AB and DE, so that DE is perpendicular to BC. (By symmetry, AB is perpendicular to EF.) Then DE bisects BC and its major arc, and likewise AB bisects EF and its major arc.

Then by the Broken Chord Theorem, EF cuts AB into parts (AB + AC)/2 and (AB - AC)/2, and by the Intersecting Chords Theorem, (AB + AC)/2 * (AB - AC)/2 = (EF/2) * (EF/2), which is equal to ( BC/2) * (BC/2). Therefore, (AB)^2 - (AC)^2 = (BC)^2, which is the Pythagorean Theorem.

--Stuart Anderson