X(3), X(4), X(20), X(1670), X(1671)
antipodes Ao, Bo, Co of A, B, C on the circumcircle
vertices of the antimedial triangle
reflections of H in A, B, C
infinite points of the McCay cubic
isogonal conjugates of the X3-OAP points, see Table 53
We meet KO++ in Special isocubics §6.5.2.
KO++ is the locus of pivots of all pK having the same asymptotic directions as the McCay cubic i.e. having three real asymptotes perpendicular to the sidelines of the Morley triangle. With the pivots X(3), X(4), X(20) we obtain the McCay cubic K003 itself, the McCay orthic cubic K049 and K096 respectively.
The homothety h(H,1/2) transforms KO++ into K026, the Musselman (first) cubic or KN++.
See also K142 for asymptotes parallel to the sidelines of the Morley triangle.
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.
See also the pencil mentioned in the page K071.