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Pivotal Isogonal Cubic


A pivotal isogonal cubic is a self-isogonal cubic that possesses a pivot point, i.e., in which points P lying on the conic and their isogonal conjugates are collinear with a fixed point Q known as the pivot of the cubic.

Pivotal isogonal cubics pass through the vertices of the Cevian triangle of the pivot point.

Let the trilinear coordinates of X be alpha:beta:gamma, then X^' has trilinear coordinates alpha^(-1):beta^(-1):gamma^(-1), or equivalently betagamma:gammaalpha:alphabeta. If the trilinear coordinates of P are x:y:z, then collinearity requires

 |x y z; alpha beta gamma; betagamma gammaalpha alphabeta|=0,

so the self-isogonal cubic with pivot point P=x:y:z has a trilinear equation of the form

 xalpha(beta^2-gamma^2)+ybeta(gamma^2-alpha^2)+zgamma(alpha^2-beta^2)=0.

The only self-isogonal triangle center is the incenter I, which a self-isogonal cubic therefore must pass through. Self-isogonal cubics also pass through the excenters J_A, J_B, and J_C.

The following table summarizes some named pivotal isogonal cubics together with their pivot points and parameters x.

Kimberling (1998, p. 240) gives lists of triangle centers passing through the pivotal isogonal cubics generated by the following pivot points: X_2, X_4, X_5, X_6, X_7, X_8, X_(10), X_(13), X_(14), X_(20), X_(27), X_(30), X_(63), X_(69), X_(92), X_(144), X_(174), X_(189), X_(226), X_(265), and X_(333).


See also

Darboux Cubic, M'Cay Cubic, Napoleon-Feuerbach Cubic, Neuberg Cubic, Orthocubic, Pivot Point, Self-Isogonal Cubic, Self-Isotomic Cubic, Thomson Cubic, Triangle Cubic

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References

Cundy, H. M. and Parry, C. F. "Some Cubic Curves Associated with a Triangle." J. Geom. 53, 41-66, 1995.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Yff, P. "Two Families of Cubics Associated with a Triangle." In MAA Notes, No. 34. Washington, DC: Math. Assoc. Amer., 1994.

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Pivotal Isogonal Cubic

Cite this as:

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

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