Ptolemy's Theorem

Let a quadrilateral ABCD be inscribed in a circle. Then prove the sum of the products of the two pairs of opposite sides equals the product of its two diagonals. In other words,

AD·BC + AB·CD = AC·BD

Discussion: There are many approaches to a proof of this important traditional geometry theorem. One approach is to construct an auxilary figure to achieve similar triangles which can then be used to set up the desired result. The following HINT is one way to approach that. Notice that angles CDB and CAB subtend the same chord CB. Therefore the angles have the same measure.

Hint: Locate point M on BD such that angle ACB = MCD.

This will allow the establishment of similar triangles DMC and ABC.

Likewise we can establish similar triangles CAD and CBM.

## Converse.

Given a quadrilateral ABCD with the sum of the products of the two pairs of opposited sides equal to the product of the two diagonals. That is, AD·BC + AB·CD = AC·BD. Prove that ABCD is inscribed in a circle.

## Extension

Let ABC be an equilateral triangle inscribed in a circle. If P is a point on the circumcircle, then of PA, PB, and PC, the larger is always the sum of the two shorter.

## Extension

Use the Theorem of Ptolemy to prove the Pythagorean theorem.

## Extension

Use the Theorem of Ptolemy and the Law of Sines to develop the additions and subtractions formulas for sines of angles.