## Pythagorean Theorem – Cool Proof

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.

Pythagorean Theorem Proof – Part I

But the Pythagorean Theorem works both ways.  If we have a right triangle labeled in the standard way then $a^2+b^2=c^2$ and if $a^2+b^2=c^2$ then we have a right triangle.  Here is a proof of the second part.

Pythagorean Theorem Proof – Part II