Ptolemy's Theorem
A sweet generaliztion of the Pythagorean Theorem, of which John D Cook reminded me in this post: johndcook.com/blog/2024/08/24/ptolemys-theorem/
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.
This applet lets you look in both directions.