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Basic Parameters of the Kimberling Center X(50)

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In a triangle , if two points have barycentric coordinates and , then the point with barycentric coordinates is called their barycentric product.

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The line known as the Brocard axis passes through the circumcenter , the symmedian point and the isodynamic points and .

The center lies on the Brocard axis and is the barycentric product of and [1].

Let

be the side lengths,

be the circumradius, inradius and semiperimeter,

and

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

Introduce the parameters , , and in Conway notation, where is the Brocard angle.

Then

,

and

.

You can drag the vertices .

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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 center" if 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 barycentric coordinates as a constant, it is called a "neutral center" (the centroid is the only "neutral center"). Conversely, a triangle center is said to be "odd center" if it is neither even nor neutral.

Standard barycentric coordinates of a point with respect to a reference triangle have a sum of 1.

Classification: even center

Reference

[1] C. Kimberling. "Encyclopedia of Triangle Centers." (Dec 13, 2022) faculty.evansville.edu/ck6/encyclopedia.

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