Feuerbach cubic

The Napoleon-Feuerbach cubic is the pivotal isogonal cubic with nine-point center N as the pivot point. It therefore has trilinear equation (α^2 - β^2) γ cos(A - B) + β(γ^2 - α^2) cos(A - C) + α(β^2 - γ^2) cos(B - C) = 0. It passes through Kimberling centers X_i for i = 1 (incenter I), 3 (circumcenter O), 4 (orthocenter H), 5 (nine-point center N), 17 (first Napoleon point), 18 (second Napoleon point), 54 (Kosnita point), 61, 62, 195, 627, 628, 2120, 2121, as well as the excenters J_A, J_B, and J_C.

pivotal isogonal cubic | triangle cubic