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Pivotal Isotomic Cubic


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

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; 1/(a^2alpha) 1/(b^2beta) 1/(c^2gamma)|=0,

so the pivotal isotomic cubic with pivot point P=x:y:z has a trilinear equation of the form

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

A pivotal isotomic cubic also passes through the triangle centroid G of the reference triangle, the vertices of the anticomplementary triangle, the vertices of the reference triangle, and the vertices of the Cevian triangle of Q.

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


See also

Lucas Cubic, Pivot Point, Self-Isotomic Cubic

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References

Yff, P. "Two Families of Cubics Associated with a Triangle." In MAA Notes, No. 34. Washington, DC: Math. Assoc. Amer., 1994.

Referenced on Wolfram|Alpha

Pivotal Isotomic Cubic

Cite this as:

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

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