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X(2), X(1645), X(1646), X(1647), X(1648), X(1649), X(1650), X(6544), X(13636), X(13722), X(14401), X(14434), X(33568), X(33569), X(33570), X(33571), X(33572), X(33573), X(41176), X(62579), X(62596), X(62611)

midpoints of ABC

infinite points of the sidelines 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.

G is an isolated point on the curve with nodal tangents passing through the infinite points of the Steiner ellipse.

K219 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.

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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).

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Locus property

Let P be a point on the Steiner inellipse. Its tangent meets the circum-parabola with perspector P at two points on K219.