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K429
equK429

X(6), X(66), X(193), X(393), X(571), X(608), X(1974), X(2911)

vertices of the orthic triangle

K429 is a nodal cubic with node K meeting the sidelines of ABC at the vertices of the orthic triangle HaHbHc.

The nodal tangents are parallel to the asymptotes of the Jerabek hyperbola. Compare K429 and K260, a very similar cubic.

The tangents at A, B, C pass through X(32).

For any point M on the Euler line, the trilinear polar L of M meets the lines KHa, KHb, KHc at Ma, Mb, Mc. The triangles ABC and MaMbMc are perspective at N and the locus of N is K429.

L envelopes the inscribed parabola with focus X(112), perspector X(648), directrix the line HK.

Since K429 is an unicursal cubic, it is very easy to find a parametrization : for any point Z = u : v : w distinct of K, the point Z' lies on K429 and its first barycentric coordinate is : [(b^2 - c^2) SA u + a^2 (SB v - SC w)] / (c^2 v - b^2 w).

K429 is also psK(X1974, X4, X6) in Pseudo-Pivotal Cubics and Poristic Triangles.