Darboux Cubic


The Darboux cubic Z(X_(20)) of a triangle DeltaABC is the locus of all pedal-cevian points (i.e., of all points whose pedal triangle is perspective with DeltaABC). It is a self-isogonal cubic with pivot point given by the de Longchamps point L (Kimberling center X_(20)). It therefore has parameter x=cosA-cosBcosC and trilinear equation


(Cundy and Parry 1995).

The Darboux cubic is symmetric with respect to the circumcenter O, so if P lies on the cubic, then so does the reflection of P through O.

It passes through Kimberling centers X_i for i=1 (incenter I), 3 (circumcenter O), 4 (orthocenter H), 20 (de Longchamps point L), 40 (Bevan point V), 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.

See also

Lucas Cubic, Self-Isogonal Cubic, Triangle Cubic

Explore with Wolfram|Alpha


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

Referenced on Wolfram|Alpha

Darboux Cubic

Cite this as:

Weisstein, Eric W. "Darboux Cubic." From MathWorld--A Wolfram Web Resource.

Subject classifications