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See the general equation at CL010

X(115), X(1648)

midpoints of ABC

points at infinity of ABC sidelines

vertices of the cevian triangle of X(115)

A1(X115) is a member of the class CL010 of cubics (Allardice cubics) and probably the most remarkable since it is a K+ with asymptotes concurrent at the point labelled E = (b^2 - c^2)^2 (b^4 + c^4 - 4 a^2 SA) : : and with perpendicular nodal tangents which are parallel to the asymptotes of the Kiepert hyperbola. These tangents are also tangent to the inscribed ellipse with center X(620), the anticomplement of X(115), with perspector the isotomic conjugate of X(115). This ellipse passes through G, X(32), X(439), X(593), X(1509) and a number of quite simple centers not mentioned in ETC. The barycentric equation of this ellipse is :

 

The anticomplement of K203 is K052.