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X(2), X(4240), X(5466), X(5468), X(6548) points at infinity of ABC sidelines (inflexion points) 

See "Tucker cubics" in the Downloads page and Special Isocubics §8.3. The Tucker nodal cubic T(G) = cK(#X2, X2) is the only unicursal Tucker cubic. It is an isotomic conicopivotal cubic with pivotal conic the Steiner ellipse. See also isotomic nK0 cubics. Compare K015 with K228, isogonal conicopivotal cubic. The complement of K015 is K219 = A1(G), an Allardice cubic (see CL010). The isogonal transform of K015 is K229. K015 and K229 are two members of CL064. The Hessian of K015 is its homothetic under h(G, 1/3). More generally, the n^{th} Hessian of K015 is the homothetic of K015 under the homothety with center G, ratio (1/3)^{n}. Locus properties :




Let M = u : v : w be a point and M' its isotomic conjugate. The polar lines of M and M' in the Steiner circumellipse intersect at f(M) = f(M') which is not defined when M is the centroid G or one of the vertices of the reference and antimedial triangles. This transformation is given by : f(M) = (v  w) / (v + w) : (w  u) / (w + u) : (u  v) / (u + v). f transforms the line at infinity and the Steiner circumellipse into K015 which gives a lot of points on K015 with rather simple coordinates. Very few of these points are listed in ETC. For example : f(X99) = f(X523) = X5468, f(X648) = f(X525) = X4240, f(X671) = f(X524) = X5466, f(X903) = f(X519) = X6548. 



Figure 1 Pc, Pi are the circumscribed and inscribed parabolas, Fc, Fi are their foci with Fi on the circumcircle (O), Dc, Di are their directrices with Di passing through H. The line M1M2 is tangent to the Steiner ellipse at M, the isotomic conjugate of the common infinite point of Pc, Pi. Recall that the parameter of Pi is 4 times that of Pc. Q077 is the locus of Fc, Q080 is the envelope of Dc, Q078 and Q079 are the loci of the vertices of Pi, Pc. Pc and Pi are obviously homothetic at a point Q which lies on the Bataille acnodal cubic K656. 

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