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Lucas Cubic


LucasCubic

The Lucas cubic is a pivotal isotomic cubic having pivot point at Kimberling center X_(69), the isogonal conjugate of the orthocenter, i.e., the locus of points P such that the Cevian triangle of P is the pedal triangle of some point Q.

The equation in trilinear coordinates is

 alphacosA(b^2beta^2-c^2gamma^2)+betacosB(c^2gamma^2-a^2alpha^2) 
 +gammacosC(a^2alpha^2-b^2beta^2)=0.

Not only is the Lucas cubic invariant under isotomic conjugate, but also under cyclocevian conjugation.

When P runs through the Lucas cubic, Q runs through the Darboux cubic.

The Lucas cubic passes through Kimberling centers X_i for i=2 (triangle centroid G), 4 (orthocenter H), 7 (Gergonne point Ge), 8 (Nagel point Na), 20 (de Longchamps point L), 69 (symmedian point of the anticomplementary triangle), 189, 253, 329, 1032, and 1034.


See also

Darboux Cubic, Pivotal Isogonal Cubic, Triangle Cubic

This entry contributed by Floor van Lamoen

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References

Gibert, B. "Lucas Cubic." http://perso.wanadoo.fr/bernard.gibert/Exemples/k007.html.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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Lucas Cubic

Cite this as:

van Lamoen, Floor. "Lucas Cubic." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/LucasCubic.html

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