Associative Binary Operations and the Pythagorean Theorem

In a recent article [2], L. Berrone presented a new approach to the Pythagorean Theorem (PT). The idea is to derive the geometric theorem from analytic and algebraic properties, by methods of functional equations. (So we are not dealing with a method that was an option for the ancients!) I thought about Berrone’s ideas, within his context of functional equations. Some pleasant surprises fell out. Then a surprising gift of functional equations back to geometry closed the circle for me – and will close this article.