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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" ยง6 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 circumstelloid passing through the cusps of the Steiner deltoid. We meet its complement in Table 11 : CPCC points related curves. K071 is also a member of the class CL063. See also CL019 and PseudoPivotal Cubics and Poristic Triangles. See a generalization and the related cubics K648, K649, K650, K651, K652. 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 circumhyperbola passing through X(20) and K071 generate a pencil of circumstelloids spK(X3, Q on the line X550X1216) 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).

