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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.
In Euclidean geometry, Brahmagupta's formula, named after the 7th century Indian mathematician, is used to find the area of any convex cyclic quadrilateral (one that can be inscribed in a circle) given the lengths of the sides. Its generalized version, Bretschneider's formula, can be used with non-cyclic quadrilateral.
A corollary to Euler's quadrilateral theorem is the inequality + + + + where equality holds if and only if the quadrilateral is a parallelogram. Euler also generalized Ptolemy's theorem, which is an equality in a cyclic quadrilateral, into an inequality for a convex quadrilateral. It states that
Japanese theorem for cyclic quadrilaterals; N. Newton's theorem (quadrilateral) ... Ptolemy's theorem This page was last edited on 2 November 2020, at 21:29 (UTC). ...
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 ...