# circumcircle

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

Such circle is called a circumcircle. Its radius is the circumradius, and its center is the circumcenter. The circumcenter lies at the intersection of the perpendicular bisectors of the sides of the triangle.

Since the perpendicular bisector of a segment is the locus of points at the same distance from the segment endpoints, 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 polygon, 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 parallelogram has no circumcircle, for no circle passes through the four vertices. A quadrilateral that does possess a circumcircle is called a cyclic quadrilateral.

 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