Basic Parameters of the Kosnita Point, Kimberling Center X(54)

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

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 [1].


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 .


Contributed by: Minh Trinh Xuan (August 25)
Open content licensed under CC BY-NC-SA


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.


[1] C. Kimberling. "Encyclopedia of Triangle Centers." (May 23, 2023)


Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.