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Eigencenter


Let T be a central triangle and let U(T) be its unary cofactor triangle. Then T and U(T) are perspective, and their perspector is called the eigencenter of T.

Let the A-, B-, and C-vertices of T be denoted x_i:y_i:z_i for i=1, 2, 3. Also define

s=y_3(x_1x_2+z_1z_2)-y_1(x_2x_3+z_2z_3)
(1)
t=z_1(x_2x_3+y_2y_3)-z_2(x_1x_3+y_1y_3)
(2)
u=z_3(x_1x_2+y_1y_2)-z_1(x_2x_3+y_2y_3)
(3)
v=y_1(x_2x_3+z_2z_3)-y_2(x_1x_3+z_1z_3).
(4)

Also define

 x=st-uv
(5)

and y and z cyclically. Then the eigencenter of T is the point x:y:z.

The following table summarizes eigencenters of named triangles that are Kimberling centers.


See also

Unary Cofactor Triangle

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References

Kimberling, C. "Glossary: A Support Page for Encyclopedia of Triangle Centers." http://faculty.evansville.edu/ck6/encyclopedia/glossary.html.

Referenced on Wolfram|Alpha

Eigencenter

Cite this as:

Weisstein, Eric W. "Eigencenter." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Eigencenter.html

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