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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.

• the Hessian of the cubic K656 also mentioned in the page K015.

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.