Basic Parameters of the Kimberling Center X(52)
Given a triangle , the Kimberling center is the orthocenter of the orthic triangle (shown in red) .[more]
The point is the centroid of the orthic triangle. The symmedian point of a triangle is the isogonal conjugate of the centroid. The nine-point center is and the circumcenter is . The points , and are collinear, as are the points , and .
, , be the side lengths,
, , be the circumradius, inradius and semiperimeter of ,
, , be the exact trilinear coordinates of with respect to and .
&LeftBracketingBar;SubscriptBox["AX", "52"]=2 SA2+(2 R2-Sω)2-S22 R,
da=a(SA(2 R2-Sω)+S2)4 R2 S,
dX52=s S+(r+R)(2 R2-Sω)2 R2.
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." (Jun 1, 2023) faculty.evansville.edu/ck6/encyclopedia.