Basic Parameters of the Kimberling Center X(55)
Given a triangle , the Kimberling center is the center of homothety of the tangential, intangents and extangents triangles .[more]
The sides of the tangential triangle are tangent to the circumcircle of at , , . See the related links for the definitions of the intangents and extangents triangles.
The point is on the line , where and are the incenter and circumcenter of .
, , be the side lengths,
, , be the circumradius, inradius and semiperimeter of ,
, , be the exact trilinear coordinates of with respect to and .
&LeftBracketingBar;AX55&RightBracketingBar;=R(b c(R+r)-R r(4 R+r))R+r,
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.