Basic Parameters of the Kosnita Point, Kimberling Center X(54)
Let be the circumcenter of the triangle and let , and be the circumcenters of the triangles , and , respectively. Then the lines , , intersect at , which is called the Kosnita point .[more]
Let , , be the exact trilinear coordinates of with respect to and set .
, , 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.
&LeftBracketingBar;AX54&RightBracketingBar; =SA2-R2 SA2 Sω-5 R2+4 R6-7 R2 S2-R4 Sω+2 S2 Sω(2 Sω-5 R2)2 ,
da =a (SA2+2 SA (2 R2-Sω)-S2)4 S (5 R2-2 Sω),
dX54 =2 r3+11 r2 R+16 r R2+4 R3-(6 r+R) s22 (5 R2-2 Sω).
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." (May 23, 2023) faculty.evansville.edu/ck6/encyclopedia.