# circumcircle

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