I was taken with a “Proof Without Words” of Ptolemy’s Theorem by William Derrick and James Hirstein in an old(Nov 2012) edition of The College Mathematics Journal. I had this vision of triangles expanding and rotating into place to form a parallelogram. What a neat proof! I wanted to do one. Since Ptolemy’s Theorem, “In an inscribed quadrilateral, the product of the diagonals is equal to the sum of the products of opposite sides” is a generalization of the Pythagorean Theorem why not try to prove Pythagorean Theorem using Derrick and Hirstein’s technique. Here is my attempt.
But the Pythagorean Theorem works both ways. If we have a right triangle labeled in the standard way then and if then we have a right triangle. Here is a proof of the second part.