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X(2), X(4), X(3146), X(11008) reflections A', B', C' of A, B, C in the sidelines of ABC images of A, B, C under h(G,4) vertices of pedal triangle of L1 X(11008) = image of X(6) under h(G,10) points at infinity of the Thomson cubic 

This cubic belongs to the pencil of cubics containing the Neuberg cubic, the Soddy cubic and the union of the three altitudes. See "Two Remarkable Pencils..." in the Downloads page. Its asymptotes are parallel to those of the Thomson cubic. See the related K127. A property by Angel Montesdeoca (20211118), see here. Let ABC be a triangle, P a point, C(P) the circumconic with perspector P and A' the pole of BC with respect to the conic tangent at A to C(P) and passing through B, C, P. The points B' and C' are defined cyclically. The triangles ABC and A'B'C' are orthologic if and only if P is on the cubic K117. The orthologic centers Q and Q', collinear with P, lie on the cubic K127. On the other hand, the triangles ABC and A'B'C' are parallelogic if and only if P is on a Tucker cubic and the parallelogic centers also lie on another Tucker cubic. Their equations are (x+y+z)(1/x+1/y+1/z) + 5/3 = 0 and (x+y+z)(1/x+1/y+1/z)  2/3 = 0 respectively. See "Tucker cubics" in the Downloads page. 
