# 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