---
tag: 0GSDHYMA1T
kind: statement
role: theorem
name: Thales's theorem
msc: [51M04]
uses: [060W5CBZP6, 0GJAJC6TNZ, 0S706JBZBG]
sources:
  - url: https://en.wikipedia.org/wiki/Thales%27s_theorem
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Thales%27s_theorem
    locator: Converse
  - url: https://en.wikipedia.org/wiki/Thales%27s_theorem
    locator: History
same_as:
  wikidata: Q285719
  wikipedia: Thales's theorem
---

## Statement

Let $A$, $B$ and $C$ be distinct points of a [[circle|060W5CBZP6]] in the Euclidean plane $\mathbb{R}^2$. Suppose that $AC$ is a diameter, so that the centre is the [[midpoint|0S706JBZBG]] of $A$ and $C$.
Then the [[angle|0GJAJC6TNZ]] $\angle ABC$ is a right angle: $\angle ABC = \pi / 2$.
Conversely, if $A$, $B$, $C$ are distinct points with $\angle ABC = \pi / 2$, then $B$ is on the circle with the diameter $AC$.

## Remarks

Thales's theorem is the special case of the inscribed angle theorem for a central angle of $\pi$.
An equivalent form: the circumcentre of a right triangle is the midpoint of its hypotenuse.
A proof: let $O$ be the centre. The triangles $OAB$ and $OBC$ are isosceles, and the angle sum of $ABC$ is $\pi$.

## History

The theorem is part of Proposition 31 of Book III of Euclid's Elements. Tradition gives it to Thales of Miletus, and sometimes to Pythagoras.

## Explanations

### L1 human

Take a diameter of a circle and a third point on the circle. Join this point to the two ends of the diameter. The angle at the third point is always a right angle.