EMAT 6690 Summer 2003

Assignment #2

Ptolemy's Theorem

M. Bauers


Click HERE for background information on Ptolemy.


Ptolemy's theorem is

The sum of the product of the two opposite sides of a cyclic quadrilateral equals the product of the diagonals.


(AD * BC) + (AB * BC) = AC *BD.


Ptolemy's Theorem provides a way to prove trigonometric identities. It implies the Pythagorean Theorem. The Pyhthagorean Theorem is utilized once the qudrilateral becomes a rectangle. Since three points determine a circle, the forth point of the qudrilateral becomes the constraint and determines whether the quadrilateral is cyclic or not. Once the constraint is met then the product of the diagonals of the cyclic quadrilateral equals the sum of the products of the opposite sides of the cyclic quadrilateral.

Proof of Ptolemy's Theorem

from cut-the-knot


When the quadrilateral is not cyclic then Ptolemy's Theorem becomes an inequality.

(AD * BC) + (AB * BC) > AC *BD