circumcircle Circumcircle 2005-02-05 16:05:30 2007-04-22 02:43:15 Definition circumcenter circumcentre circumradius circle triangle center radius \usepackage{graphicx} %\usepackage{xypic} \usepackage{bbm} \newcommand{\Z}{\mathbbmss{Z}} \newcommand{\C}{\mathbbmss{C}} \newcommand{\R}{\mathbbmss{R}} \newcommand{\Q}{\mathbbmss{Q}} \newcommand{\mathbb}[1]{\mathbbmss{#1}} \newcommand{\figura}[1]{\begin{center}\includegraphics{#1}\end{center}} \newcommand{\figuraex}[2]{\begin{center}\includegraphics[#2]{#1}\end{center}} \newtheorem{dfn}{Definition} For any triangle $ABC$ there is always a circle passing through its three vertices. \begin{center} \includegraphics{circumcircle} \end{center} Such circle is called a \emph{circumcircle}. Its radius is the \emph{circumradius}, and its center is the \emph{circumcenter}. The circumcenter lies at the intersection of the perpendicular bisectors of the sides of the triangle. {\small 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 $C$.} \bigskip 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.