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# M'Cay Cubic

The M'Cay cubic is a self-isogonal cubic given by the locus of all points whose pedal circle touches the nine-point circle, or equivalently, the locus of all points for which , the isogonal conjugate of , and the circumcenter of a reference triangle are collinear, where the equivalence follows from one of the Fontené theorems.

Its pivot point is the circumcenter (Kimberling center ), so it has parameter and trilinear equation

(Gallatly 1913, p. 80; Cundy and Parry 1995).

The M'Cay cubic of a triangle passes through Kimberling centers for (incenter ), 3 (circumcenter ), 4 (orthocenter ), 1075, 1745, and excenters , , and of , but is omitted from Kimberling's list of pivotal isogonal cubics (Kimberling 1998, p. 240).

The M'Cay cubic is the locus of points for which the pedal and circumcevian triangles are perspective (and in fact, even homothetic).

Pivotal Isogonal Cubic, Triangle Cubic

## References

Cundy, H. M. and Parry, C. F. "Some Cubic Curves Associated with a Triangle." J. Geom. 53, 41-66, 1995.Gallatly, W. "M'Cay's Cubic." §109 in The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, p. 80, 1913.Gibert, B. "McCay Cubic = Griffiths Cubic." http://perso.wanadoo.fr/bernard.gibert/Exemples/k003.html.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.

M'Cay Cubic

## Cite this as:

Weisstein, Eric W. "M'Cay Cubic." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MCayCubic.html