The Neuberg cubic Z(X_30) of a triangle Δ A B C is the locus of all points P whose reflections in the sidelines B C, C A, and A Bform a triangle perspective to Δ A B C. It is a self-isogonal cubic with pivot point at the Euler infinity point X_30, so it has parameter x = cos A - 2cos B cos C and trilinear equation (α^2 - β^2) γ(cos C - 2cos A cos B) + (γ^2 - α^2) β(cos B - 2cos A cos C) + (β^2 - γ^2) α(cos A - 2cos B cos C) (Cundy and Parry 1995).

self-isogonal cubic | triangle cubic