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# Ptolemy’s theorem

If $ABCD$ is a cyclic quadrilateral, then the product of the two diagonals is equal to the sum of the products of opposite sides.

$AC\cdot BD=AB\cdot CD+AD\cdot BC.$ |

When the quadrilateral is not cyclic we have the following inequality

$AB\cdot CD+BC\cdot AD>AC\cdot BD$ |

An interesting particular case is when both $AC$ and $BD$ are diameters, since we get another proof of Pythagoras’ theorem.

Keywords:

Quadrilateral, Circle, Cyclic, Ptolemy

Related:

CyclicQuadrilateral, ProofOfPtolemysTheorem, PtolemysTheorem, PythagorasTheorem, CrossedQuadrilateral

Type of Math Object:

Theorem

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

51-00*no label found*60K25

*no label found*18-00

*no label found*68Q70

*no label found*37B15

*no label found*18-02

*no label found*18B20

*no label found*

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new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz

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new correction: Misspelled name by DavidSteinsaltz

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new correction: underline-typo by Filipe

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new correction: cocycle pro cocyle by pahio

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new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier

new image: expected waiting time by robert_dodier

new image: plot W(t) = P(waiting time <= t) by robert_dodier

## Comments

## Ptolemy's thm

Maybe the word "Ptolemy" could go in the keywords; the search

engine currently finds no hits on that word.

We might mention some of the easy corollaries of Pt's thm. E.g.

if we specialize to the case in which $AC$ and

$BD$ are diameters of the given circle, we get Pythagoras's theorem.

LH