Thales's theorem

Builds on Circle Angle at a point Midpoint

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.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Angle at a point concept
Circle concept
Midpoint concept
Thales's theorem statement

Tag	`0GSDHYMA1T`
URI	`m2://0GSDHYMA1T`
Kind	statement, theorem
Writing	W1 stated
MSC	51M04
Status	This copy of the site has no build data. Run `m2 status 0GSDHYMA1T`.
Uses	Circle, Angle at a point, Midpoint
Same as	Wikidata Q285719 Wikipedia Thales's theorem

Thales's theorem

Statement

Remarks

History

Reading path

Further reading

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