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  "body": "\n## Definition\n\nLet $(X, d)$ be a [[metric space|0VKD1TP7F3]] and let $S \\subseteq X$ be a set of points.\nThe set $S$ is *cospherical* if there are a point $c \\in X$ and a real number $r$ such that $d(x, c) = r$ for every $x \\in S$.\nThat is, $S$ is contained in the [[sphere|P9CM4GQ2T0]] with centre $c$ and radius $r$.\n\n## Remarks\n\nThis is the form of Mathlib's `EuclideanGeometry.Cospherical`.\nIn the Euclidean plane, cospherical points are the same as [[concyclic points|049DCJZ448]].\nIn $\\R^n$, any $n + 1$ affinely independent points are cospherical, and their sphere is the [[circumsphere|MRC4WMVTAH]] of the simplex they span.\n\nExample: the eight vertices $(\\pm 1, \\pm 1, \\pm 1)$ of a cube in $\\R^3$ are cospherical.\nEach lies at distance $\\sqrt 3$ from $c = (0, 0, 0)$.\n\nNon-example: the points $(0, 0)$, $(1, 0)$, $(2, 0)$ of $\\R^2$ are not cospherical.\nA point equidistant from the first two has $x$-coordinate $1/2$, and a point equidistant from the last two has $x$-coordinate $3/2$, so no centre exists.\n",
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