For any triangleMathworldPlanetmath ABC there is always a circle passing through its three vertices.

Such circle is called a circumcircleMathworldPlanetmath. Its radius is the circumradius, and its center is the circumcenter. The circumcenter lies at the intersectionMathworldPlanetmath of the perpendicular bisectorsMathworldPlanetmath of the sides of the triangle.

Since the perpendicular bisector of a segment is the locus of points at the same distanceMathworldPlanetmath from the segment endpointsMathworldPlanetmath, the points on the perpendicular bisector of AB are equidistant to A and B. The points in the perpendicular bisector of BC are equidistant to B and C, and thus the intersection point O is at the same distance from A,B and C.

In a more general setting, if P is any polygonMathworldPlanetmathPlanetmath, its circumcircle would be a circle passing through all vertices, and circumradius and circumcenter are defined similarly. However, unlike triangles, circumcircles need not to exist for any polygon. For instance, a non-rectangular parallelogramMathworldPlanetmath has no circumcircle, for no circle passes through the four vertices. A quadrilateralMathworldPlanetmath that does possess a circumcircle is called a cyclic quadrilateralMathworldPlanetmath.

Title circumcircle
Canonical name Circumcircle
Date of creation 2013-03-22 15:00:32
Last modified on 2013-03-22 15:00:32
Owner yark (2760)
Last modified by yark (2760)
Numerical id 8
Author yark (2760)
Entry type Definition
Classification msc 51-00
Related topic Triangle
Related topic CyclicQuadrilateral
Related topic SimsonsLine
Defines circumcenter
Defines circumcentre
Defines circumradius