---
tag: 058YS05YP5
kind: statement
role: theorem
name: Ptolemy's theorem
msc: [51M04]
uses: [0RCP929Y1M, 049DCJZ448, 0NPPGHQNXQ]
sources:
  - url: https://en.wikipedia.org/wiki/Ptolemy%27s_theorem
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Ptolemy%27s_theorem
    locator: Ptolemy's inequality
same_as:
  wikidata: Q459547
  wikipedia: Ptolemy's theorem
---

## Statement

Let $A$, $B$, $C$ and $D$ be [[concyclic points|049DCJZ448]] of the Euclidean plane $\mathbb{R}^2$, and let $d$ be the [[Euclidean distance|0NPPGHQNXQ]]. Suppose that the segments $AC$ and $BD$ meet in a point that is inside both segments. Thus $ABCD$ is a convex [[cyclic quadrilateral|0RCP929Y1M]] with the diagonals $AC$ and $BD$.
Then
$$d(A, C)\, d(B, D) = d(A, B)\, d(C, D) + d(B, C)\, d(A, D).$$

## Remarks

In words: the product of the diagonals is the sum of the products of the opposite sides.
The converse is also true. If a quadrilateral has this property, then it is cyclic.
For each four points of the plane, $d(A, C)\, d(B, D) \leq d(A, B)\, d(C, D) + d(B, C)\, d(A, D)$. This is Ptolemy's inequality.
For a rectangle with the sides $a$ and $b$, the theorem gives $a^2 + b^2 = c^2$, with the diagonal $c$. This is the [[Pythagorean theorem|0VP3A1290K]].

## History

The theorem has the name of the astronomer Ptolemy. He used it to make his table of chords, a trigonometric table for astronomy.

## Explanations

### L1 human

Put four points in order on a circle and join them into a quadrilateral. Multiply the two diagonals. You get the same number as when you multiply the opposite sides in pairs and add the two products.