X(4), X(5), X(20), X(76), X(5562)
cusps of the Steiner deltoid of ABC
its contacts with the altitudes of ABC and with the sidelines of ABC (traces of X(69) on the sidelines of ABC)
infinite points of the McCay cubic
points of pK(X6, X5562) on the circumcircle
We meet this cubic in the FG paper "The Lemoine cubic and its generalizations" §6 under the name Kconc (see the Downloads page).
This cubic is a K60+ with three real asymptotes parallel to those of the McCay cubic and concurring at Z5 = X(5891), the reflection of X(51) about X(5). K071 is actually the only circum-stelloid passing through the cusps of the Steiner deltoid.
The isogonal transform of K071 is psK(X25 x X97, X97, X3) = spK(X5562, X1216), a CircumNormal cubic as in Table 25.
K071 is spK(X3, X1216) as in CL055.
The union of the line at infinity with the rectangular circum-hyperbola passing through X(20) and K071 generate a pencil of circum-stelloids spK(X3, Q on the line X550-X1216) which contains K080, X268, K827, K928.
All these stelloids have their asymptotes parallel to those of the McCay cubic K003 and concurring on the line X(3)X(64). The remaining common points of these stelloids are X(4), X(20), X(15318).