Basic Parameters of the Kimberling Center X(57)
Given a triangle , the Kimberling center is the homothetic center of and the orthic triangle of the intouch triangle (or contact triangle) .[more]
The vertices of the orthic triangle are the feet of the altitudes of .
The vertices of the intouch triangle are the points of tangency of and its incircle.
The point is on line , where and are the incenter and circumcenter of .
, , be the side lengths,
, , be the circumradius, inradius and semiperimeter of ,
, , be the exradius values,
, , be the exact trilinear coordinates of with respect to and .
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 3, 2023) faculty.evansville.edu/ck6/encyclopedia.