X(3), X(4), X(6), X(20), X(253), X(1350)
antipodes of A, B, C on the circumcircle
points at infinity of the Thomson cubic
Let PaPbPc be the pedal triangle of point P. The parallel at P to BC meets the parallels at Pa to PB and PC at Ab and Ac respectively. Define Bc, Ba, Ca, Cb likewise.
Consider Ga, Gb, Gc centroids of triangles AAbAc, BBcBa, CCaCb. Triangles PaPbPc and GaGbGc are perspective if and only if P lies on the Antreas cubic.
This cubic is a central circum-K++ cubic with center O. It is spK(X2, X376) in CL055.
Its isogonal transform is K615.
The Darboux cubic K004 and the decomposed cubic which is the union of the circumcircle and the Euler line generate a pencil of central cubics with center O passing through H, X(20) and the reflections A', B', C' of A, B, C about O. Each cubic has the same asymptotic directions as one of the isogonal pK of the Euler pencil. This pencil also contains (apart K004) the cubics K047, K080, K426, K443, K566 corresponding to K002, K003, pK(X6, X3146), K006, K005 respectively. These cubics are those in the column P = [X20] of Table 54.