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X(3), X(4), X(20), X(54) reflections of A, B, C about O infinite points of the Napoleon cubic K005 

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. K566 is that corresponding to the Napoleon cubic K005. It is spK(X5, X548) in CL055. See also Table 54 and Table 58. This pencil also contains (apart K004) the cubics K047, K080, K426, K443 corresponding to K002, K003, pK(X6, X3146), K006 respectively. These cubics are those in the column P = [X20] of Table 54. The isogonal transform of K566 is K848 = spK(X20, X548). 
