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). Ptolemy used the theorem as an aid to creating his table of chords, a trigonometric table that he applied to astronomy.

If the quadrilateral is given with its four vertices *A*, *B*, *C*, and *D* in order, then the theorem states that:

where the vertical lines denote the lengths of the line segments between the named vertices.

This relation may be verbally expressed as follows:

*If a quadrilateral is inscribable in a circle then the product of the measures of its diagonals is equal to the sum of the products of the measures of the pairs of opposite sides.*

Moreover, the converse of Ptolemy's theorem is also true:

*In a quadrilateral, if the sum of the products of its two pairs of opposite sides is equal to the product of its diagonals, then the quadrilateral can be inscribed in a circle.*

Read more about Ptolemy's Theorem: Corollaries, Ptolemy's Inequality

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### Famous quotes containing the word theorem:

“To insure the adoration of a *theorem* for any length of time, faith is not enough, a police force is needed as well.”

—Albert Camus (1913–1960)