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In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral (a quadrilateral whose vertices lie on a common circle). The theorem is named after the Greek astronomer and mathematician Ptolemy (Claudius Ptolemaeus). [ 1 ]
Ptolemy's theorem expresses the product of the lengths of the two diagonals e and f of a cyclic quadrilateral as equal to the sum of the products of opposite sides: [9]: p.25 [2] e f = a c + b d , {\displaystyle \displaystyle ef=ac+bd,}
By Ptolemy's theorem, if a quadrilateral is given by the pairwise distances between its four vertices A, B, C, and D in order, then it is cyclic if and only if the product of the diagonals equals the sum of the products of opposite sides: = +.
Ptolemy's inequality is often stated for a special case, in which the four points are the vertices of a convex quadrilateral, given in cyclic order. [2] [3] However, the theorem applies more generally to any four points; it is not required that the quadrilateral they form be convex, simple, or even planar.
Japanese theorem for cyclic polygons; Japanese theorem for cyclic quadrilaterals; Kosnita's theorem; Lester's theorem; Milne-Thomson circle theorem; Miquel's theorem; Monge's theorem; Mrs. Miniver's problem; Pivot theorem; Pizza theorem; Squaring the circle; Poncelet's porism; Ptolemy's theorem; Ptolemy's table of chords; Regiomontanus' angle ...
Euler also generalized Ptolemy's theorem, which is an equality in a cyclic quadrilateral, into an inequality for a convex quadrilateral. It states that + where there is equality if and only if the quadrilateral is cyclic. [24]: p.128–129 This is often called Ptolemy's inequality.
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Japanese theorem for cyclic quadrilaterals; N. ... Pitot theorem; Ptolemy's theorem This page was last edited on 2 November 2020, at 21:29 (UTC). ...
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