Basic Parameters of the Circumcenter

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Given a triangle , let , , be the midpoints of the sides , , . The three lines through , , and perpendicular to the sides of intersect at the circumcenter. The center is the Kimberling center [1].


Let , , be the exact trilinear coordinates of the point with respect to and let .

Let , , be the side lengths opposite the corresponding vertices and let , , , be the circumradius, inradius, exradius for and semiperimeter of , respectively. Let .

When , , are in Conway notation, then




You can drag the vertices , and .


Contributed by: Minh Trinh Xuan (August 2022)
Open content licensed under CC BY-NC-SA



A triangle center is said to be even when its barycentric coordinates can be expressed as a function of three variables , , that all occur with even exponents. If the center of a triangle has constant barycentric coordinates, it is called a neutral center (the centroid is the only neutral center). A triangle center is said to be odd if it is neither even nor neutral.

Standard barycentric coordinates of a point with respect to a reference triangle are normalized to a sum of 1.


[1] C. Kimberling. "Encyclopedia of Triangle Centers." (Aug 2, 2022)

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