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  "body": "\n## Definition\n\nLet $ABC$ be a triangle in the Euclidean plane $\\mathbb{R}^2$, with [[circumcentre|0F98FZK14A]] $O$ and circumradius $R = d(O, A)$, where $d$ is the Euclidean distance.\nThe circumcircle of $ABC$ is the [[circle|060W5CBZP6]] with the centre $O$ and the radius $R$:\n$$\\{\\, X \\in \\mathbb{R}^2 \\;:\\; d(X, O) = R \\,\\}.$$\nIt is the unique circle that contains $A$, $B$ and $C$.\n\n## Remarks\n\nFor a triangle in $\\mathbb{R}^n$, the circumcircle is the circle in the plane of the triangle with the centre $O$ and the radius $R$. Mathlib's `Affine.Simplex.circumsphere` is the sphere with this centre and radius in the ambient space.\nThe [[law of sines|0PASQ34GHG]] gives $2R = a / \\sin \\alpha$, with the side $a$ opposite the angle $\\alpha$.\nA polygon whose vertices all lie on one circle is a cyclic polygon. Each triangle is cyclic, but not each quadrilateral is.\n\n## Explanations\n\n### L1 human\n\nThrough any three points that are not on one line, there is exactly one circle. For a triangle, this is the circumcircle. For the right triangle with the vertices $(0, 0)$, $(4, 0)$ and $(0, 3)$, the circumcircle has the centre $(2, 1.5)$ and the radius $2.5$.\n",
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