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
Its asymptotes are parallel to those of the Thomson cubic.
See the related K127.
A property by Angel Montesdeoca (2021-11-18), see here.
Let ABC be a triangle, P a point, C(P) the circum-conic 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.