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