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Unary Cofactor Triangle


Let P_i=x_i:y_i:z_i be trilinear points for i=1, 2, 3. The A-vertex of the unary cofactor triangle is then defined as the point

 y_2z_3-z_2y_3:z_2x_3-x_2z_3:x_2y_3-y_2x_3,

and the B-vertex and C-vertex are defined cyclically.

The following table summarizes the unary cofactor triangles for common named triangles.

The vertices are the isogonal conjugates of the vertices of the line-polar triangle of the points P_i.

If T is a triangle and U(T) is its unary cofactor triangle, then U(U(T))=T, and T and U(T) are perspective, with the perspector being known as the eigencenter.

A triangle is perspective to DeltaABC iff its unary cofactor triangle is perspective to ABC. Also, triangle T_1 circumscribes triangle T_2 iff U(T_2) circumscribes U(T_1) (Kimberling and van Lamoen 1999).


See also

Eigencenter

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References

Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Kimberling, C. "Glossary: A Support Page for Encyclopedia of Triangle Centers." http://faculty.evansville.edu/ck6/encyclopedia/glossary.html.Kimberling, C. and van Lamoen, F. M. "Central Triangles." Nieuw Arch. Wisk. 17, 1-20, 1999.

Referenced on Wolfram|Alpha

Unary Cofactor Triangle

Cite this as:

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

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