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  "body": "\n## Definition\n\nLet $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean space $\\mathbb{R}^n$, and let $d$ be the [[Euclidean distance|0NPPGHQNXQ]]. Let $P$ be the plane through $A$, $B$ and $C$, their [[affine span|0YMB1676R1]].\nThe circumcentre of $ABC$ is the unique point $O$ of $P$ with\n$$d(O, A) = d(O, B) = d(O, C).$$\nThis common distance $R$ is the circumradius of $ABC$.\n\n## Remarks\n\nThe point $O$ exists and is unique. In the plane, it is the common point of the three [[perpendicular bisectors|055VNYT0PB]] of the sides. Two of them meet in one point because the sides are not parallel.\nThe circumcentre is inside the triangle if the triangle is acute, on the longest side if it is right, and outside if it is obtuse.\nIn a right triangle, the circumcentre is the midpoint of the hypotenuse, by [[Thales's theorem|0GSDHYMA1T]].\nMathlib defines the circumcentre of each simplex in the same way, inside the affine span of the vertices.\n\n## Explanations\n\n### L1 human\n\nThe circumcentre is the point that is equally far from the three corners of a triangle. To find it, draw the perpendicular bisectors of two sides; they cross at the circumcentre. A circle with this centre through one corner passes through all three.\n",
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