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# Circumcenter

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

 (1)

and the exact trilinear coordinates are therefore

 (2)

where is the circumradius, or equivalently

 (3)

The circumcenter is Kimberling center .

The distance between the incenter and circumcenter is , where is the circumradius and is the inradius.

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

 (4) (5) (6) (7) (8) (9) (10) (11) (12)

where is the triangle triangle centroid, is the orthocenter, is the incenter, is the symmedian point, is the nine-point center, is the Nagel point, is the de Longchamps point, is the circumradius, is Conway triangle notation, and 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,

 (13)

where is the midpoint of side , is the circumradius, and 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.

 triangle Kimberling circumcenter anticomplementary triangle orthocenter circumcircle mid-arc triangle circumcenter circum-medial triangle circumcenter circumnormal triangle circumcenter circum-orthic triangle circumcenter circumtangential triangle circumcenter contact triangle incenter D-triangle midpoint of and Euler triangle nine-point center excentral triangle Bevan point extouch triangle circumcenter of extouch triangle Feuerbach triangle nine-point center first Brocard triangle midpoint of Brocard diameter first Morley triangle first Morley center first Yff circles triangle internal similitude center of the circumcircle and incircle Fuhrmann triangle Fuhrmann center hexyl triangle incenter inner Napoleon triangle triangle centroid inner Vecten triangle complement of Lucas tangents triangle isogonal conjugate of medial triangle nine-point center mid-arc triangle incenter orthic triangle nine-point center outer Napoleon triangle triangle centroid outer Vecten triangle complement of reference triangle circumcenter reflection triangle -Ceva conjugate of second Brocard triangle midpoint of Brocard diameter second Yff circles triangle external similitude center of the circumcircle and incircle Stammler triangle circumcenter tangential triangle circumcenter of the tangential triangle

The circumcenter and orthocenter are isogonal conjugates.

The orthocenter of the pedal triangle formed by the circumcenter concurs with the circumcenter 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.

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

## References

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." http://www.ics.uci.edu/~eppstein/junkyard/circumcenter.html.Johnson, 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." http://faculty.evansville.edu/ck6/tcenters/class/ccenter.html.Kimberling, C. "Encyclopedia of Triangle Centers: X(3)=Circumcenter." http://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X3.

Circumcenter

## Cite this as:

Weisstein, Eric W. "Circumcenter." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Circumcenter.html