X(2), X(1645), X(1646), X(1647), X(1648), X(1649), X(1650)
midpoints of ABC
K219 is the Allardice cubic A1(G). It is a singular cubic with singularity G. It is tangent to the sidelines of ABC at the midpoints.
It is also :
• the complement of the Tucker nodal cubic K015 = A2(G).
• the locus of roots of all tridents of the class CL029.
• the locus of tripolar centroids of the points on the line at infinity. See a generalization at CL045.
Two conics, one inscribed and one circumscribed, with the same center P have parallel asymptotes if and only if P lies on K219 or on the line at infinity. If "asymptotes" is replaced with "axes", we obtain the Thomson cubic K002.
See here for other related properties (Angel Montesdeoca, 2019-07-13).