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).
