Basic Parameters of the Kimberling Center X(61)
Given a triangle , construct the three exterior equilateral triangles on its sides with centers , , . Then the lines , , intersect at the first Napoleon point (see related links).[more]
Let be the circumcenter and be the symmedian point. These points determine the Brocard axis (shown in red).
Then the Kimberling center is the isogonal conjugate of , which is on the Brocard axis .
, , be the side lengths,
, , be the circumradius, inradius and semiperimeter of ,
, , be the exact trilinear coordinates of with respect to and .
Introduce the parameters , , and in Conway notation, where is the Brocard angle.
You can drag the vertices , and .[less]
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 have a sum of 1.
 C. Kimberling. "Encyclopedia of Triangle Centers." (Jul 19, 2023) faculty.evansville.edu/ck6/encyclopedia.