Ptolemy's theorem PtolemysTheorem 2001-10-06 03:49:39 2004-03-15 15:58:00 Theorem Quadrilateral Circle Cyclic Ptolemy \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{xypic} If $ABCD$ is a cyclic quadrilateral, then the product of the two diagonals is equal to the sum of the products of opposite sides. \begin{center} \includegraphics{ptolemy} \end{center} \[AC\cdot BD = AB\cdot CD + AD \cdot BC.\] 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.