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Basic Parameters of the Kimberling Center X(55)

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Given a triangle , the Kimberling center is the center of homothety of the tangential, intangents and extangents triangles [1].

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The sides of the tangential triangle are tangent to the circumcircle of at , , . See the related links for the definitions of the intangents and extangents triangles.

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

Let

, , be the side lengths,

, , be the circumradius, inradius and semiperimeter of ,

,

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

Then

&LeftBracketingBar;AX55&RightBracketingBar;=R(b c(R+r)-R r(4 R+r))R+r,

da=a(s-a)2(R+r),

dX55=r(4 R+r)R+r.

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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