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K447

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

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 K447, its isogonal transform K446 and K448, an axial cubic.

The singular focus of K447 is the reflection of O in X(74).

The tangents at A, B, C to K447 are the symmedians.

K447 is also psK(X74, X1494, X3) in Pseudo-Pivotal Cubics and Poristic Triangles.

See also Table 46.