M'Cay Cubic


The M'Cay cubic Z(X_3) 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 P for which P, the isogonal conjugate P^' of P, and the circumcenter O of a reference triangle DeltaABC are collinear, where the equivalence follows from one of the Fontené theorems.

Its pivot point is the circumcenter O (Kimberling center X_3), so it has parameter x=cosA and trilinear equation


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

The M'Cay cubic of a triangle DeltaABC passes through Kimberling centers X_i for i=1 (incenter I), 3 (circumcenter O), 4 (orthocenter H), 1075, 1745, and excenters J_A, J_B, and J_C of DeltaABC, but is omitted from Kimberling's list of pivotal isogonal cubics (Kimberling 1998, p. 240).

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

See also

Pivotal Isogonal Cubic, Triangle Cubic

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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.", C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.

Referenced on Wolfram|Alpha

M'Cay Cubic

Cite this as:

Weisstein, Eric W. "M'Cay Cubic." From MathWorld--A Wolfram Web Resource.

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