In any geometryMathworldPlanetmath where a circle is defined, a collectionMathworldPlanetmath of points are said to be concyclicMathworldPlanetmath if there is a circle that is incidentMathworldPlanetmathPlanetmathPlanetmath with all the points.

Remarks. Suppose all points being considered below lie in a Euclidean planeMathworldPlanetmath.

  • Any two points P,Q are concyclic. In fact, there are infinitely many circles that are incident to both P and Q. If PQ, then the pencil 𝔓 of circles incident with P and Q share the property that their centers are collinearMathworldPlanetmath. It is easy to see that any point on the perpendicular bisectorMathworldPlanetmath of PQ¯ serves as the center of a unique circle in 𝔓.

  • Any three non-collinear points P,Q,R are concyclic to a unique circle c. From the three points, take any two perpendicular bisectors, say of PQ¯ and PR¯. Then their intersectionMathworldPlanetmath O is the center of c, whose radius is |OP|.

  • Four distinct points A,B,C,D are concyclic iff CAD=CBD.

Title concyclic
Canonical name Concyclic
Date of creation 2013-03-22 16:07:58
Last modified on 2013-03-22 16:07:58
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 7
Author CWoo (3771)
Entry type Definition
Classification msc 51-00