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.

Title	Ptolemy’s theorem
Canonical name	PtolemysTheorem
Date of creation	2013-03-22 11:43:13
Last modified on	2013-03-22 11:43:13
Owner	drini (3)
Last modified by	drini (3)
Numerical id	18
Author	drini (3)
Entry type	Theorem
Classification	msc 51-00
Classification	msc 60K25
Classification	msc 18-00
Classification	msc 68Q70
Classification	msc 37B15
Classification	msc 18-02
Classification	msc 18B20
Related topic	CyclicQuadrilateral
Related topic	ProofOfPtolemysTheorem
Related topic	PtolemysTheorem
Related topic	PythagorasTheorem
Related topic	CrossedQuadrilateral