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  "body": "\n## Statement\n\nLet $A$, $B$ 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 $C$.\nThen the [[angle|0GJAJC6TNZ]] $\\angle ABC$ is a right angle: $\\angle ABC = \\pi / 2$.\nConversely, if $A$, $B$, $C$ are distinct points with $\\angle ABC = \\pi / 2$, then $B$ is on the circle with the diameter $AC$.\n\n## Remarks\n\nThales's theorem is the special case of the inscribed angle theorem for a central angle of $\\pi$.\nAn equivalent form: the circumcentre of a right triangle is the midpoint of its hypotenuse.\nA proof: let $O$ be the centre. The triangles $OAB$ and $OBC$ are isosceles, and the angle sum of $ABC$ is $\\pi$.\n\n## History\n\nThe theorem is part of Proposition 31 of Book III of Euclid's Elements. Tradition gives it to Thales of Miletus, and sometimes to Pythagoras.\n\n## Explanations\n\n### L1 human\n\nTake 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.\n",
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