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  "body": "\n## Definition\n\nLet $S$ be a set of points of the Euclidean space $\\mathbb{R}^n$, and let $d$ be the [[Euclidean distance|0NPPGHQNXQ]].\nThe points of $S$ are concyclic if they lie on one [[circle|060W5CBZP6]]. That is, there are a [[plane|0V3XAMF3C1]] $P$ that contains $S$, a point $c \\in P$ and a real number $r \\geq 0$ with $d(x, c) = r$ for each $x \\in S$.\n\n## Remarks\n\nThree points that are not collinear are always concyclic. Thus each triangle has a circumcircle.\nFour or more points of the plane are not concyclic in general. Four concyclic points are the vertices of a cyclic quadrilateral.\nThe centre $c$ is on the perpendicular bisector of each two distinct points of $S$.\nMathlib's `EuclideanGeometry.Concyclic` asks for points that are cospherical and coplanar. This is the same condition.\n\n## Explanations\n\n### L1 human\n\nConcyclic points are points on one common circle. The four corners of a rectangle are concyclic: the circle has its centre at the crossing of the diagonals. The four points $(0, 0)$, $(1, 0)$, $(2, 0)$ and $(0, 1)$ are not concyclic, because a circle meets a line in at most two points.\n",
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