Ptolemy's Theorem

$\begin{figure}\begin{center}\BoxedEPSF{Ptolemys_Theorem.epsf}\end{center}\end{figure}$

If a Quadrilateral is inscribed in a circle (i.e., for a cyclic quadrilateral), the sum of the products of the two pairs of opposite sides equals the product of the diagonals

$\begin{displaymath} AB\times CD+BC\times DA=AC\times BD. \end{displaymath}$

This fact can be used to derive the Trigonometry addition formulas.

See also Fuhrmann's Theorem, Ptolemy Inequality

References

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 42-43, 1967.