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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.

K429 is the barycentric products X(3) × K620, X(4) × K009, X(6) × K555, X(25) × K257, X(186) × K724.

K429 is the locus of poles of pKs meeting the circumcircle at the same points as the orthocubic K006. The locus of the pivots and isopivots are K028 and K389 respectively.