# Basic Parameters of the Kimberling Center X(52)

Initializing live version

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

Given a triangle , the Kimberling center is the orthocenter of the orthic triangle (shown in red) [1].

[more]

The point is the centroid of the orthic triangle. The symmedian point of a triangle is the isogonal conjugate of the centroid. The nine-point center is and the circumcenter is . The points , and are collinear, as are the points , and .

Let

, , be the side lengths,

,

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

Then

&LeftBracketingBar;SubscriptBox["AX", "52"]=2 SA2+(2 R2-Sω)2-S22 R,

da=a(SA(2 R2-Sω)+S2)4 R2 S,

dX52=s S+(r+R)(2 R2-Sω)2 R2.

You can drag the vertices , and .

[less]

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." (Jun 1, 2023) faculty.evansville.edu/ck6/encyclopedia.

## Permanent Citation

Minh Trinh Xuan

 Feedback (field required) Email (field required) Name Occupation Organization Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback. Send