Ptolemy’s theorem

Draw a quadrilateral by pick four arbitrary points on a circle and connecting them cyclically.

Now multiply the lengths of the pairs of opposite sides. In the diagram below this means multiplying the lengths of the two horizontal-ish blue sides and the two vertical-ish orange sides.

Ptolemy’s theorem says that the sum of the two products described above equals the product of the diagonals.

To put it in colorful terms, the product of the blue sides plus the product of the orange sides equals the product of the green diagonals.

The converse of Ptolemy’s theorem also holds. If the relationship above holds for a quadrilateral, then the quadrilateral can be inscribed in a circle.

Note that if the quadrilateral in Ptolemy’s theorem is a rectangle, then the theorem reduces to the Pythagorean theorem.

Related posts

• Area of a quadrilateral as a determinant
• Van Aubel’s theorem
• Ptolemy and trig tables