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
[[File:McCayCubicLocus.png|thumb|

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

Equation of the McCay cubic
The equation of the McCay cubic in barycentric coordinates $$x:y:z$$ is
 * $$\sum_{\text{cyclic}}(a^2(b^2+c^2-a^2)x(c^2y^2-b^2z^2))=0.$$

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
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.