Ptolemy's theorem

For a cyclic quadrilateral, the product of the diagonal lengths equals the sum of the products of the two pairs of opposite side lengths.

Cyclic quadrilateral: A quadrilateral whose four vertices lie on one circle. The theorem applies precisely when all four vertices lie on a common circle.
Ptolemy's identity for a rectangle: The relation that a rectangle's diagonal squared equals the sum of its two side lengths squared. A rectangle is cyclic, so the theorem reduces directly to the familiar diagonal formula.
Ptolemy: A Greek mathematician and astronomer of the second century CE, known for the Almagest. The theorem bears his name and appears in his mathematical work.
Quadrilateral: A polygon with four sides and four vertices. The theorem concerns a particular class of quadrilaterals with four vertices on a circle.
Brahmagupta's formula: A formula for the area of a cyclic quadrilateral in terms of its four side lengths and semiperimeter. It also depends on cyclicity but determines area rather than relating diagonals and sides.
Ptolemy's inequality: For any convex quadrilateral, the product of the diagonals is at most the sum of the opposite-side products. Equality holds exactly for cyclic quadrilaterals, extending the theorem beyond its stated case.
Ptolemy's inequality in quadrilateral geometry: The use of Ptolemy's inequality to bound diagonal lengths in convex quadrilaterals. The theorem supplies the equality case for broader diagonal-length estimates.
Almagest: Ptolemy's second-century treatise on mathematical astronomy and the motion of celestial bodies. Its chord tables and geometric arguments include the relation now called Ptolemy's theorem.
Circumcircle: A circle passing through every vertex of a polygon. Its existence is the defining geometric condition in Ptolemy's theorem.
Bretschneider's formula: A formula for the area of a general quadrilateral using its sides and the sum of a pair of opposite angles. Unlike Ptolemy's theorem, it applies beyond cyclic quadrilaterals and concerns area.

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