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.

Cundy, H. M. and Parry, C. F. "Some Cubic Curves Associated with a Triangle." J. Geom. 53, 41-66, 1995.

Gibert, B. "Darboux Cubic." http://perso.wanadoo.fr/bernard.gibert/Exemples/k004.html.

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

Rubio, P. "Cubic Lines Relative to a Triangle." J. Geom. 34, 152-171, 1989.

CITE THIS AS:

Weisstein, Eric W. "Darboux Cubic." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/DarbouxCubic.html