The Darboux cubic Z(X_20) of a triangle Δ A B C is the locus of all pedal-cevian points (i.e., of all points whose pedal triangle is perspective with Δ A B C). 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 = cos A - cos B cos C and trilinear equation (cos A - cos B cos C) α(β^2 - γ^2) + (cos B - cos C cos A) β(γ^2 - α^2) + (cos C - cos A cos B) γ(α^2 - β^2) = 0 (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.

Lucas cubic | self-isogonal cubic | triangle cubic