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.

Explanations

L1 human

Through any three points that are not on one line, there is exactly one circle. For a triangle, this is the circumcircle. For the right triangle with the vertices $(0, 0)$, $(4, 0)$ and $(0, 3)$, the circumcircle has the centre $(2, 1.5)$ and the radius $2.5$.