The Lucas cubic is a pivotal isotomic cubic having pivot point at Kimberling
center 
,
the isogonal conjugate of the orthocenter,
i.e., the locus of points 
such that the Cevian triangle
of 
is the pedal
triangle of some point 
.
The equation in trilinear coordinates is
 |
Not only is the Lucas cubic invariant under isotomic conjugate, but also under cyclocevian conjugation.
When 
runs through the Lucas cubic, 
runs through the Darboux cubic.
The Lucas cubic passes through Kimberling centers 
for 
(triangle centroid 
), 4 (orthocenter 
), 7 (Gergonne
point 
),
8 (Nagel point 
), 20 (de Longchamps point 
), 69 (symmedian
point of the anticomplementary triangle),
189, 253, 329, 1032, and 1034.
