circumcircle

For any triangle $ABC$ there is alwaysa 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 intersectionof the perpendicular bisectors of the sides of the triangle.

Since the perpendicular bisector of a segmentis 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 circumcircleis 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