McCay cubic

In Euclidean geometry, the McCay cubic (also called M'Cay cubic or Griffiths cubic) is a cubic plane curve in the plane of a reference triangle and associated with it. It is the third cubic curve in Bernard Gilbert's Catalogue of Triangle Cubics and it is assigned the identification number K003.

Definition

Template:Legend-line Template:Legend-line Template:Legend-line Template:Legend-line Template:Legend-line

The McCay cubic can be defined by locus properties in several ways. For example, the McCay cubic is the locus of a point Template:Mvar such that the pedal circle of Template:Mvar is tangent to the nine-point circle of the reference triangle Template:Math. The McCay cubic can also be defined as the locus of point Template:Mvar such that the circumcevian triangle of Template:Mvar and Template:Math are orthologic.

Equation of the McCay cubic

The equation of the McCay cubic in barycentric coordinates $x:y:z$ is

$\sum _{\text{cyclic}}(a^2(b^2+c^2-a^2)x(c^2{y}^2-b^2{z}^2))=0.$

The equation in trilinear coordinates $\alpha :\beta :\gamma$ is

$\alpha ({\beta }^2-{\gamma }^2)\cos A+\beta ({\gamma }^2-{\alpha }^2)\cos B+\gamma ({\alpha }^2-{\beta }^2)\cos C=0$

McCay cubic as a stelloid

McCay cubic with its three concurring asymptotes

A stelloid is a cubic that has three real concurring asymptotes making 60° angles with one another. McCay cubic is a stelloid in which the three asymptotes concur at the centroid of triangle ABC. A circum-stelloid having the same asymptotic directions as those of McCay cubic and concurring at a certain (finite) is called McCay stelloid. The point where the asymptoptes concur is called the "radial center" of the stelloid. Given a finite point X there is one and only one McCay stelloid with X as the radial center.