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Basic Parameters of the Kimberling Center X(47)

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In the triangle , let be the incenter, be the orthocenter, be the Schiffler point, be the -beth conjugate of (see the glossary at [1]) and be the -Ceva conjugate of . Then the center is the intersection of the lines and .

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Let

, , be the side lengths,

, , be the circumradius, inradius and semiperimeter of and

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

Then

,

and

.

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 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: odd center

Reference

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

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