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
and the exact trilinear coordinates
is the circumradius, or equivalently
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
is the triangle triangle centroid, is the orthocenter, is the incenter, is the symmedian point,
is the nine-point
is the Nagel point, is the de Longchamps point,
is the circumradius,
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,
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.
and orthocenter are isogonal conjugates.
The orthocenter of the pedal
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
See alsoBrocard Diameter
, Carnot's Theorem
, Euler's Inequality
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ReferencesCarr, 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
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.
Cite this as:
Weisstein, Eric W. "Circumcenter." From
MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Circumcenter.html