(Mathematics - Geometry)

A circumcenter is the center of a circle known as the circumcircle which circumscribes a triangle. It is found at the intersection point of the three perpendicular bisectors of a triangle, and also lies on the Euler line.

Given triangle ABC, the perpendicular bisectors are concurrent at the circumcenter.

Let line l be the perpendicular bisector of AB, let line m be the perpendicular bisector of BC, and let D be the intersection point of l and m. Since a point is on a perpendicular bisector of two points if and only if that point is equidistant from the pair of points, it follows that since D is on l, |D - A| = |D - B| and since D is on m, |D - B| = |D - C|. Consequently, |D - A| = |D - C|. D is equidistant from A and C, therefore D is on the perpendicular bisector of AC. All three lines share D and so the lines are concurrent at D. D is equidistant from all three corners. Hence D is the circumcenter.

Since the circumcenter is equidistant from A, B and C, it must lie on both lines l and m. Lines l and m cannot be paralell by the definition of triangle, so the two lines intersect at one point only, proving the uniqueness of the circumcenter.

See also other triangle centers: incenter, centroid, orthocenter.