Circumcircle

Definition

Let $ABC$ be a triangle in the Euclidean plane $\mathbb{R}^2$ , with circumcentre $O$ and circumradius $R = d(O, A)$ , where $d$ is the Euclidean distance. The circumcircle of $ABC$ is the circle with the centre $O$ and the radius $R$ : $\{\, X \in \mathbb{R}^2 \;:\; d(X, O) = R \,\}.$ It is the unique circle that contains $A$ , $B$ and $C$ .

Remarks

For a triangle in $\mathbb{R}^n$ , the circumcircle is the circle in the plane of the triangle with the centre $O$ and the radius $R$ . Mathlib's Affine.Simplex.circumsphere is the sphere with this centre and radius in the ambient space. The law of sines gives $2R = a / \sin \alpha$ , with the side $a$ opposite the angle $\alpha$ . A polygon whose vertices all lie on one circle is a cyclic polygon. Each triangle is cyclic, but not each quadrilateral is.

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Circle concept
Circumcentre concept
Circumcircle definition

Tag	`0RKB5MKC05`
URI	`m2://0RKB5MKC05`
Kind	definition
Writing	W1 stated
MSC	51M04
Defines	Circumcircle
Uses	Circumcentre, Circle

Circumcircle

Definition

Remarks

Reading path

Further reading

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