Editing Ptolemy's theorem (section)

{{short description|Relates the 4 sides and 2 diagonals of a quadrilateral with vertices on a common circle}}
[[File:Ptolemy equality.svg|right|thumb|upright=1.25|Ptolemy's theorem is a relation among these lengths in a cyclic quadrilateral.<math>\definecolor{V}{RGB}{148,0,211} \definecolor{B}{RGB}{0,0,255} \definecolor{R}{RGB}{204,0,0}
{\color{V}AC}\cdot{\color{V}BD}={\color{B}AB}\cdot{\color{B}CD}+{\color{R}BC}\cdot{\color{R}AD}</math>]]

In [[Euclidean geometry]], '''Ptolemy's theorem''' is a relation between the four sides and two diagonals of a [[cyclic quadrilateral]] (a quadrilateral whose [[Vertex (geometry)#Of a polytope|vertices]] lie on a common circle). The theorem is named after the [[Roman Greece|Greek]] [[astronomer]] and [[mathematician]] [[Ptolemy]] (Claudius Ptolemaeus).<ref>C. Ptolemy, [[Almagest]], Book 1, Chapter 10.</ref> Ptolemy used the theorem as an aid to creating [[Ptolemy's table of chords|his table of chords]], a trigonometric table that he applied to astronomy.

If the vertices of the cyclic quadrilateral are ''A'', ''B'', ''C'', and ''D'' in order, then the theorem states that:

: <math>AC\cdot BD = AB\cdot CD+BC\cdot AD</math>

This relation may be verbally expressed as follows:

:''If a quadrilateral is [[Cyclic quadrilateral|cyclic]] then the product of the lengths of its diagonals is equal to the sum of the products of the lengths of the pairs of opposite sides.''

Moreover, the [[Theorem#Converse|converse]] of Ptolemy's theorem is also true:

:''In a quadrilateral, if the sum of the products of the lengths of its two pairs of opposite sides is equal to the product of the lengths of its diagonals, then the quadrilateral can be inscribed in a circle i.e. it is a [[cyclic quadrilateral]].''

To appreciate the utility and general significance of Ptolemy’s Theorem, it is especially useful to study its main [[#Corollaries|Corollaries]].