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Basic Parameters of the Kimberling Center X(62)

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Given a triangle , construct the three interior equilateral triangles on the sides of with centers , , . Then the lines , , intersect at the second Napoleon point (see related links).

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Let be the circumcenter and be the symmedian point. These points determine the Brocard axis (shown in red).

Then the Kimberling center is the isogonal conjugate of and is on the Brocard axis [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 , , and in Conway notation, where is the Brocard angle.

Then

,

,

.

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." (Jul 20, 2023) faculty.evansville.edu/ck6/encyclopedia.

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