 Ptolemy's theorem

For four vertices in cyclic order on a circle, the product of the diagonal lengths is the sum of the products of opposite side lengths. One proof uses the cross-ratio with second point mapped to zero: its negative sign gives $|r|+1=|1-r|$, which becomes the displayed length relation.

Ptolemy's theorem is a fundamental result in geometry that applies to cyclic quadrilaterals — that is, quadrilaterals whose vertices lie on the circumference of a circle.