X(3), X(76), X(3224)
vertices of the first Brocard triangle,
midpoints of ABC
infinite points of K410
points on the Steiner ellipse and K003
points on the circumcircle and K184
See Tucker cubics.
K512 is the isogonal transform of K444.
K512 is also psK(X2, X2, X3) in Pseudo-Pivotal Cubics and Poristic Triangles.
K512 is a bicentric cubic : if M = u : v : w is a point lying on K512 then the points M1 = w : u : v and M2 = v : w : u also lie on K512.
This is actually true for any psK(X2, X2, M) with M ≠ X2.