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Basic Parameters of the Kimberling Center X36

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Given a triangle , let be its circumcenter. Then the Kosnita triangle is defined to have its vertices at the circumcenters of the circumcircles of , and .

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By Kosnita's theorem, , and are concurrent at a point, hence called the Kosnita point. The reflection of the Kosnita triangle in is called the Trinh triangle.

The Kimberling center is the homothetic center of the intangents triangle (purple) and the Trinh triangle (orange) [1].

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 , , , in Conway notation, where is the Brocard angle.

Then

,

,

.

You can drag the vertices , and .

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Contributed by: Minh Trinh Xuan (January 2023)
Open content licensed under CC BY-NC-SA

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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." (Oct 18, 2022) faculty.evansville.edu/ck6/encyclopedia.

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