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Basic Parameters of the Kimberling Center X(57)

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Given a triangle , the Kimberling center is the homothetic center of and the orthic triangle of the intouch triangle (or contact triangle) [1].

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The vertices of the orthic triangle are the feet of the altitudes of .

The vertices of the intouch triangle are the points of tangency of and its incircle.

The point is on line , where and are the incenter and circumcenter of .

Let

, , be the side lengths,

, , be the circumradius, inradius and semiperimeter of ,

,

, , be the exradius values,

, , 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." (Jul 3, 2023) faculty.evansville.edu/ck6/encyclopedia.

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