around 150 CE

### Ptolemy's Quadrilateral Theorem

The Pythagorean theorem via cyclic quadrilaterals

Claudius Ptolemy's famous work the *Almagest* gives a result on cyclic quadrilaterals, which gives the Pythagorean theorem as a special case.

Book 1, Chapter 10 of the *Almagest* contains a relation between the sides and diagonals of a cyclic quadrilateral known as Ptolemy's theorem: *AB*×*CD* + *BC*×*DA* = *AC*×*BD*. Taking the special case of the quadrilateral being a rectangle, this result reduces to the Pythagorean theorem, thus providing one of the world's first alternate proofs.