Ptolemy inequality

Depends on: AA similarity, Triangle inequality a + b > c, Opposite angles of a cyclic quadrilateral are supplementary.

Statement

Let $ABCD$ be any convex quadrilateral in the plane (not necessarily inscribed in a circle), with diagonals $AC$ and $BD$. Then there is always an inequality between the three products

and equality holds if and only if $A$, $B$, $C$, $D$ are concyclic (i.e. $ABCD$ is a cyclic quadrilateral).

Read alongside Ptolemy's theorem: that equation $AC \cdot BD = AB \cdot CD + BC \cdot AD$ holds only for cyclic quadrilaterals; drop the "inscribed in a circle" constraint and equality slackens to inequality — this is the Ptolemy inequality.