The Darboux cubic 
of a triangle 
is the locus of all pedal-cevian points (i.e., of all points
whose pedal triangle is perspective with 
). It is a self-isogonal
cubic with pivot point given by the de Longchamps
point 
(Kimberling center 
). It therefore has parameter 
and trilinear equation
 |
(Cundy and Parry 1995).
The Darboux cubic is symmetric with respect to the circumcenter 
, so if 
lies on the cubic, then so does the reflection
of 
through 
.
It passes through Kimberling centers 
for 
(incenter 
), 3 (circumcenter 
), 4 (orthocenter 
), 20 (de Longchamps point 
), 40 (Bevan
point 
),
64 (the isogonal conjugate of the de
Longchamps point), 84 (the isogonal conjugate
of the Bevan point) (Kimberling 1998, p. 240),
1490, 1498, 2130, and 2131.
