For a quadrilateral inscribed in a circle, the product of the lengths of the diagonals equals the sum of the products of the lengths of opposite sides.
We have AC × BD = AB × CD + AD × BC.
Pythagoras' theorem is in fact a special case of Ptolemy's theorem: take a rectangle with side lengths a and b and diagonal length c. It is indeed a cyclic quadrilateral, and Ptolemy's theorem states that c × c = a × a + b × b. That is the Pythagorean relation!
Ptolemy proved his theorem using the classical geometry of similar triangles and the inscribed-angle theorem, which states that angles inscribed in a circle and subtending the same arc are equal (see Le Cercle (The Circle), Bibliothèque Tangente 36, 2009, and Les Angles (Angles), Bibliothèque Tangente 53, 2015). Here, we will obtain it as the equality case of an inequality, first using the complex plane: the usual Euclidean plane in which each point is represented by its complex coordinate (a complex number representing its coordinates).
