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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $\\Delta$ be an $n$-[[simplex|YX1DJK2QAG]] in $\\R^N$ with vertices $v_0, \\ldots, v_n$.\nLet $S$ be the [[affine span|0FFAE5P6R4]] of the vertices.\nThere is exactly one point $o \\in S$ with $\\norm{o - v_0} = \\norm{o - v_1} = \\cdots = \\norm{o - v_n}$.\nThis point is the *circumcentre* of $\\Delta$, and this common distance $r$ is its *circumradius*.\nThe *circumsphere* of $\\Delta$ is the [[sphere|P9CM4GQ2T0]] $\\set{x \\in S : \\norm{x - o} = r}$ in $S$.\n\n## Remarks\n\nThe point $o$ exists and is unique because the conditions $\\norm{o - v_i}^2 = \\norm{o - v_0}^2$ for $i = 1, \\ldots, n$ are $n$ affine equations in $S$.\nTheir linear parts $v_i - v_0$ are linearly independent, by affine independence of the vertices.\n\nExample: the triangle with vertices $(0, 0)$, $(2, 0)$ and $(0, 2)$ has circumcentre $(1, 1)$ and circumradius $\\sqrt{2}$.\nEach vertex is at distance $\\sqrt{1 + 1} = \\sqrt{2}$ from $(1, 1)$.\nFor $n = 2$ this is the [[circumcircle|03VZJE6C8C]].\n\nNon-example: for four points $(0, 0)$, $(2, 0)$, $(0, 2)$ and $(3, 3)$ in the plane, there is no circle through all of them.\nThe only circle through the first three has centre $(1, 1)$, and $(3, 3)$ is at distance $2\\sqrt{2}$ from it.\nThe definition needs the vertices of a simplex, which are affinely independent.\n\nWithout the condition $o \\in S$, the centre would not be unique when $n \u003c N$: every point of the affine subspace through $o$ orthogonal to $S$ is equidistant from the vertices.\n",
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