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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

points Ua, Ub, Uc mentioned in the Neuberg cubic page. See also table 16 and table 18.

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.

Compare K080 and K307, the locus of poles of all pKs having the same asymptotic directions as the McCay cubic.

K080 is a central K60++ with center O. It is the isogonal transform of K405. It is spK(X3, X550) in CL055.

The homothety h(H,1/2) transforms KO++ into K026, the Musselman (first) cubic or KN++. Equivalently, KO++ is the anticomplement of K026.

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.