X(6), X(50), X(67), X(524), X(1989), X(1990)
K381 is a conico-pivotal isocubic with pivotal conic the parabola with focus X(2709), the inverse of X(115) in the circumcircle, and directrix the line X(5)X(6). This parabola is inscribed in the tangential triangle i.e. the anticevian triangle of K. Note that X(2709) lies on the circumcircle of this triangle.
It has node the Lemoine point K and the nodal tangents are perpendicular and parallel to the asymptotes of the Jerabek hyperbola. Its pole is X(32), the 3rd power point. Its root is X(523), the infinite point of the orthic axis hence K381 meets the sidelines of ABC on the line passing through the centers of the Kiepert and Jerabek hyperbolas.
K381 is the locus of M such that the midpoint of M and its X(32)-isoconjugate M* lies on the line GK.
The line MM* is tangent at T to the pivotal parabola. The harmonic conjugate S of T with respect to M and M* lies on the cubic and S* (also on the cubic) is the common tangential of M and M*. Furthermore, the lines KM and KM* meet the Jerabek hyperbola again at two points which are antipodes on the Jerabek hyperbola.
K381 is the isogonal transform of the isotomic cK(#X2, X850) and the X(31)-transform of the isogonal cK(#X1, X1577).
See the related cubic K185 = cK(#X2, X523).
K381 is also Kc2(X6) in CL053.
K381 is the locus of poles of circular pKs whose orthic line passes through the circumcenter of ABC. The corresponding pivots lie on the Jerabek hyperbola or on the line at infinity.
A line passing through K meets K018 again at two points P1, P2 which lie on a circum-conic passing through G. The barycentric product P1 x P2 lies on K381.