Basic Parameters of the Kimberling Center X(61)

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Given a triangle , construct the three exterior equilateral triangles on its sides with centers , , . Then the lines , , intersect at the first 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 , which is on the Brocard axis [1].

Let

, , be the side lengths,

, , 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 19, 2023) faculty.evansville.edu/ck6/encyclopedia.

Permanent Citation

Minh Trinh Xuan

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