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X(3), X(4), X(30), X(74), X(133), X(1511)

Ma, Mb, Mc : midpoints of ABC

A' on the perpendicular bisector of BC and on the parallel at A to the Euler line, B' and C' similarly

Let us consider the two following decomposed cubics : one is the union of the line at infinity and the Jerabek hyperbola, the other is the union of the circumcircle of ABC and the Euler line. Each one is clearly the isogonal transform of the other.

These cubics generate a pencil of circular circum-cubics passing through O, H, X(30), X(74). This pencil is stable under isogonal conjugation and contains the Neuberg cubic K001 (the only self-isogonal pK) and K187 (the only self-isogonal nK). The singular foci lie on the line O-X(74)-X(110)-etc and two isogonal conjugate cubics have their respective foci inverse in the circumcircle. The orthic line is the Euler line.

This pencil also contains K446, its isogonal transform K447 and K448, an axial cubic.

The singular focus of K446 is X(12041), the midpoint of OX(74).

The tangents at A, B, C to K446 concur at X(1495), the midpoint of X(23) and X(110), a point on the asymptote of the cubic. This asymptote meets K446 again at X(1511), the midpoint of OX(110).

The anticomplement of K446 is K449.

K446 is also psK(X1495, X2, X3) in Pseudo-Pivotal Cubics and Poristic Triangles.

See also Table 46, Table 50 and the related K1095.

K449 is the locus of pivots of circular pKs that pass through X(30) i.e. that have a real asymptote and an orthic line parallel to the Euler line. The locus of their poles is K1348 = psK(X1495, X2, X6) and the locus of their isopivots is K446.