but it cannot be unreasonable to presume that the astronomers, architects and construction engineers of ancient Egypt may have had some knowledge of it. Ptolemy's inequality The equation in Ptolemy's theorem is never true with non-cyclic quadrilaterals. Ptolemy's inequality is an extension of this fact, and it is a more general form of Ptolemy's theorem. It states that, given a quadrilateral ABCD, then where equality holds if and only if the quadrilateral is cyclic. This special case is equivalent to Ptolemy's theorem. Related theorem about the ratio of the diagonals Ptolemy's theorem gives the product of the diagonals (of a cyclic quadrilateral) knowing the sides, the following theorem yields the same for the ratio of the diagonals. Proof: It is known that the area of a triangle inscribed in a circle of diameter is: Writing the area of the quadrilateral as sum of two triangles sharing the same circumscribing circle, we obtain two relations for each decomposition. Equating, we obtain the announced formula. Consequence: Knowing both the product and the ratio of the diagonals, we deduce their immediate expressions: See also Casey's theorem Greek mathematics Notes References Coxeter, H. S. M. and S. L. Greitzer (1967) "Ptolemy's Theorem and its Extensions." §2.6 in Geometry Revisited, Mathematical Association of America pp. 42–43. Copernicus (1543) De Revolutionibus Orbium Coelestium, English translation found in On the Shoulders of Giants (2002) edited by Stephen Hawking, Penguin Books Amarasinghe, G. W. I. S. (2013) A Concise Elementary Proof for the Ptolemy's Theorem, Global Journal of Advanced Research on Classical and Modern Geometries (GJARCMG) 2(1): 20–25 (pdf). External links Proof of Ptolemy's Theorem for Cyclic Quadrilateral MathPages – On Ptolemy's Theorem Ptolemy's Theorem at cut-the-knot Compound angle proof at cut-the-knot Ptolemy's Theorem on PlanetMath Ptolemy Inequality on MathWorld De Revolutionibus Orbium Coelestium at Harvard. Deep Secrets: The Great Pyramid, the Golden Ratio and the Royal Cubit Ptolemy's Theorem by Jay Warendorff, The Wolfram Demonstrations Project. Book XIII of Euclid's Elements Theorems about quadrilaterals and circles Theorem Articles containing proofs Euclidean plane geometry