# 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 |