## Ptolemy's Theorem

### Stefan Gössner

*Department of Mechanical Engineering, University of Applied Sciences, Dortmund, Germany.*

**Keywords**: Ptolemy's theorem; vectorial proof; cyclic quadrilateral; geometry;

## Introduction

In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral (a quadrilateral whose vertices lie on a common circle) [1].

With given side and diagonal lengths, Ptolemy's theorem of a cyclic quadrilateral states:

For a

cyclic quadrilateralthe product of the diagonal lengths is equal to the sum of the product of the length of the pairs of opposite sides.

## Vectorial Proof

The quadrilateral can be described by a loop closure of side vectors

Expressing the diagonal vector

Both angles in the triangle corners opposite to diagonal

Resolving for the dot products

On our way for isolating

Herein the right hand side can be simplified by resolving the brackets and reordering

Otherwise expressing the diagonal vector

Both angles in the triangle corners opposite to diagonal

Now multiplying equations (4) and (5), reducing the fractions and applying the square root finally results in (1)

## References

- [1] Ptolemy's theorem, WikiPedia.
- [2] A collection of proofs of Ptolemy’s Theorem, Chaitanya's Random Pages.
- [3] Ptolemy's Theorem, Wolfram MathWorld.
- [4] Cross Product Considered Harmful, S. Goessner.