Pythagoras' from the Star of David
The following proof of the Pythagorean theorem is due to Stuart Anderson (September 17, 2010).
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.
Therefore, AB² - AC² = BC², which is the Pythagorean Theorem.