Thales's theorem

Statement

Let $A$, $B$ and $C$ be distinct points of a circle in the Euclidean plane $\mathbb{R}^2$. Suppose that $AC$ is a diameter, so that the centre is the midpoint of $A$ and $C$. Then the angle $\angle ABC$ 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.