Circumcentre

Builds on Triangle Affine span Euclidean distance Perpendicular bisector

Definition

Let $ABC$ be a triangle in the Euclidean space $\mathbb{R}^n$ , and let $d$ be the Euclidean distance. Let $P$ be the plane through $A$ , $B$ and $C$ , their affine span. The circumcentre of $ABC$ is the unique point $O$ of $P$ with $d(O, A) = d(O, B) = d(O, C).$ This common distance $R$ is the circumradius of $ABC$ .

Remarks

The point $O$ exists and is unique. In the plane, it is the common point of the three perpendicular bisectors of the sides. Two of them meet in one point because the sides are not parallel. The circumcentre is inside the triangle if the triangle is acute, on the longest side if it is right, and outside if it is obtuse. In a right triangle, the circumcentre is the midpoint of the hypotenuse, by Thales's theorem. Mathlib defines the circumcentre of each simplex in the same way, inside the affine span of the vertices.

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Affine span concept
Euclidean distance concept
Perpendicular bisector concept
Triangle concept
Circumcentre definition

Tag	`0TP2QKJ8DE`
URI	`m2://0TP2QKJ8DE`
Kind	definition
Writing	W0 stub
MSC	51M04
Defines	Circumcentre
Uses	Triangle, Affine span, Euclidean distance, Perpendicular bisector

Circumcentre

Definition

Remarks

Reading path

Further reading

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