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