McCay cubic explained

In Euclidean geometry, the McCay cubic (also called M'Cay cubic[1] or Griffiths cubic[2]) 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.[2]

Definition

The McCay cubic can be defined by locus properties in several ways.[2] For example, the McCay cubic is the locus of a point such that the pedal circle of is tangent to the nine-point circle of the reference triangle .[3] The McCay cubic can also be defined as the locus of point such that the circumcevian triangle of and are orthologic.

Equation of the McCay cubic

x:y:z

is

\sumcyclic(a2(b2+c2-a2)x(c2y2-b2z2))=0.

The equation in trilinear coordinates

\alpha:\beta:\gamma

is

\alpha(\beta2-\gamma2)\cosA+\beta(\gamma2-\alpha2)\cosB+\gamma(\alpha2-\beta2)\cosC=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.[2] 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.[4] Given a finite point X there is one and only one McCay stelloid with X as the radial center.

Notes and References

  1. Web site: Weisstein, Eric W . M'Cay Cubic . MathWorld-A Wolfram Web Resource . Wolfram Research, Inc. . 5 December 2021.
  2. Web site: Bernard Gilbert . K003 McCay Cubic = Griffiths Cubic . Cubics in the Triangle Plane . Bernard Gilbert . 5 December 2021.
  3. Book: John Griffiths . Mathematical Questions and Solutions from the Educational Times 2 (1902) 109, and 3 (1903) 29.
  4. Web site: Bernard Gilbert . McCay Stelloids . Catalogue of Triangle Cubics . Bernard Gilbert . 25 December 2021.