around 150 CE
Ptolemy's Quadrilateral Theorem
The Pythagorean theorem via cyclic quadrilaterals
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.