PlanetMath: Ptolemy's theorem

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Ptolemy's theorem

(Theorem)

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

$\includegraphics{ptolemy}$

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

When the quadrilateral is not cyclic we have the following inequality

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

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

"Ptolemy's theorem" is owned by drini. [ owner history (1) ]

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Keywords:

Quadrilateral, Circle, Cyclic, Ptolemy

Attachments:: proof of Ptolemy's theorem (Proof) by drini
proof of Ptolemy's inequality (Proof) by mathwizard

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Cross-references: Pythagoras theorem, proof, diameters, inequality, quadrilateral, opposite sides, sum, diagonals, product, cyclic quadrilateral
There is 1 reference to this entry.

This is version 6 of Ptolemy's theorem, born on 2001-10-06, modified 2004-03-15.
Object id is 100, canonical name is PtolemysTheorem.
Accessed 11780 times total.

Classification:

51-00 (Geometry :: General reference works )

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