Ptolemeus theorem

In every convex quadrangle inscribed in a circle, the product of the lengths of the diagonals is equal to the sum of the products of the lengths of its opposite sides. It is called after Claudius Ptolemeus (2nd century), who used it to deduce certain trigonometric relations.

Comments

One also encounters Ptolemy's theorem. Ptolemy's theorem is equivalent to the formula for $\sin(\alpha+\beta)$.

The converse to Ptolemy's theorem is also true. If in a convex quadrangle the product of the lengths of the diagonals is equal to the sum of the products of the lengths of its opposite sides, the four vertices are lying on a circle.

References

[a1]	H.S.M. Coxeter, S.L. Greitzer, "Geometry revisited" , Math. Assoc. Amer. (1967)
[a2]	E.A. Maxwell, "Geometry by transformations" , Cambridge Univ. Press (1975)
[a3]	M. Berger, "Geometry" , 1–2 , Springer (1987) (Translated from French)