We prove Ptolemy's theorem, and its converse, using complex numbers. More precisely, we show that for a quadrilateral ABCD, |AB||CD| + |BC||AD| = |AC||BD| holds if and only if ABCD is a cyclic quadrilateral.

A quadrilateral is cyclic if and only if opposite angles sum to 180°:
https://en.wikipedia.org/wiki/Cyclic_...

The proof here is inspired by Hahn's proof in this book:
Hahn, L., 1994. Complex numbers and geometry. Cambridge University Press.

00:00 Ptolemy's theorem
01:06 Ptolemy's inequality
03:56 Equality in Ptolemy's inequality
06:21 Strategy for the rest of the proof
07:13 Equality if and only if...
09:35 Relating arguments to angles
12:34 Conclusion