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

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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].

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Let , , be the exact trilinear coordinates of with respect to and set .

Let

, , 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.

Then

&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 .

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

Details

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.

Reference

[1] C. Kimberling. "Encyclopedia of Triangle Centers." (May 23, 2023) faculty.evansville.edu/ck6/encyclopedia.

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