A sweet generaliztion of the Pythagorean Theorem, of which John D Cook reminded me in this post: [url=https://www.johndcook.com/blog/2024/08/24/ptolemys-theorem/]johndcook.com/blog/2024/08/24/ptolemys-theorem/[/url][br][br]Multiply the lengths of opposite sides of a quadrilateral and sum the products. They are equal to the product of the diagonals if and only if the quadrilateral is cyclic. [br][br]This applet lets you look in both directions.