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Basic Parameters of the Kimberling Center X(65)

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Let and be the incenter and circumcenter of the triangle . These points determine the OI line (shown in red). The Euler lines of the four triangles , , , intersect at the Schiffler point (see related links).

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Then the Kimberling center is the isogonal conjugate of , on the OI line and is the orthocenter of the contact triangle (shown in purple) [1].

Let

, , be the side lengths,

, , be the circumradius, inradius and semiperimeter of ,

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

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." (Aug 2, 2023) faculty.evansville.edu/ck6/encyclopedia.

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