Concyclic points

Builds on Circle Plane Euclidean distance

Definition

Let $S$ be a set of points of the Euclidean space $\mathbb{R}^n$ , and let $d$ be the Euclidean distance. The points of $S$ are concyclic if they lie on one circle. That is, there are a plane $P$ that contains $S$ , a point $c \in P$ and a real number $r \geq 0$ with $d(x, c) = r$ for each $x \in S$ .

Remarks

Three points that are not collinear are always concyclic. Thus each triangle has a circumcircle. Four or more points of the plane are not concyclic in general. Four concyclic points are the vertices of a cyclic quadrilateral. The centre $c$ is on the perpendicular bisector of each two distinct points of $S$ . Mathlib's EuclideanGeometry.Concyclic asks for points that are cospherical and coplanar. This is the same condition.

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Circle concept
Euclidean distance concept
Plane concept
Concyclic points definition

Tag	`0JWCX0TW2S`
URI	`m2://0JWCX0TW2S`
Kind	definition
Writing	W1 stated
MSC	51M04
Defines	Concyclic points
Uses	Circle, Plane, Euclidean distance

Concyclic points

Definition

Remarks

Reading path

Further reading

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