The circumcenter is the center O of a triangle's circumcircle. It can be found as the intersection of the perpendicular bisectors. The trilinear coordinates of the circumcenter are


and the exact trilinear coordinates are therefore


where R is the circumradius, or equivalently


The circumcenter is Kimberling center X_3.

The distance between the incenter and circumcenter is sqrt(R(R-2r)), where R is the circumradius and r is the inradius.

Distances to a number of other named triangle centers are given by


where G is the triangle triangle centroid, H is the orthocenter, I is the incenter, K is the symmedian point, N is the nine-point center, Na is the Nagel point, L is the de Longchamps point, R is the circumradius, S_omega is Conway triangle notation, and Delta is the triangle area.

If the triangle is acute, the circumcenter is in the interior of the triangle. In a right triangle, the circumcenter is the midpoint of the hypotenuse.

For an acute triangle,


where M_i is the midpoint of side A_i, R is the circumradius, and r is the inradius (Johnson 1929, p. 190).

Given an interior point, the distances to the polygon vertices are equal iff this point is the circumcenter. The circumcenter lies on the Brocard axis.

The following table summarizes the circumcenters for named triangles that are Kimberling centers.


The circumcenter O and orthocenter H are isogonal conjugates.


The orthocenter H_(DeltaO_1O_2O_3) of the pedal triangle DeltaO_1O_2O_3 formed by the circumcenter O concurs with the circumcenter O itself, as illustrated above.

The circumcenter also lies on the Brocard axis and Euler line. It is the center of the circumcircle, second Brocard circle, and second Droz-Farny circle and lies on the Brocard circle and Lester circle. It also lies on the Jerabek hyperbola and the Darboux cubic, M'Cay cubic, Neuberg cubic, orthocubic, and Thomson cubic.

The complement of the circumcenter is the nine-point center.

See also

Brocard Diameter, Carnot's Theorem, Circle, Circumcenter of Mass, Circumcircle, Euler Line, Euler's Inequality, Euler Triangle Formula, Incenter, Lester Circle, Orthocenter, Triangle Centroid

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Carr, G. S. Formulas and Theorems in Pure Mathematics, 2nd ed. New York: Chelsea, p. 623, 1970.Dixon, R. Mathographics. New York: Dover, p. 55, 1991.Eppstein, D. "Circumcenters of Triangles.", R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929.Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163-187, 1994.Kimberling, C. "Circumcenter.", C. "Encyclopedia of Triangle Centers: X(3)=Circumcenter."

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Cite this as:

Weisstein, Eric W. "Circumcenter." From MathWorld--A Wolfram Web Resource.

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