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X(2), X(4), X(7), X(8), X(20), X(69), X(189), X(253), X(329), X(1032), X(1034), vertices of the 2nd Conway triangle, see ETC X(9776) vertices of antimedial triangle points at infinity of the Thomson cubic foci of the Steiner circumellipse (see below) CPCC or Hcevian points, see Table 11, and their isotomic conjugates seven central cyclocevian points, see Table 24. See also Table 28 : cevian and anticevian points. 

The Lucas cubic is the isotomic pK with pivot X(69), isotomic conjugate of H. It is the Thomson cubic of the antimedial triangle. See K878 for a generalization and other related cubics. Its isogonal transform is K172. It is a member of the classes CL023, CL024 of cubics. See also Table 15. It is anharmonically equivalent to the Thomson cubic. See Table 21. K007 meets the circumcircle at A, B, C and three other points on the rectangular hyperbola through X(2), X(20), X(54), X(69), X(110), X(2574), X(2575), X(2979). Locus properties :




Transformations preserving the Lucas cubic The Lucas cubic is invariant under isotomic conjugation and under several other transformations. Its is indeed the locus of point P such that :




The Lucas cubic and the foci of the Steiner ellipse The Lucas cubic is the most familiar example of isocubic which contains the (four) foci of the Steiner (circum) ellipse. See also K347 and K348. Indeed, these foci are the anticomplements of those of the inscribed Steiner ellipse which lie on the Thomson cubic since they are two isogonal conjugates collinear with the pivot G. Let then be F1, F2 the real foci and F1', F2' the imaginary foci of the Steiner ellipse. Their isotomic conjugates tF1, tF2, tF1', tF2' obviously also lie on the Lucas cubic. Furthermore :
Naturally, all these results can easily be adapted to the imaginary foci. 

More informations in Wilson's page. See the related cubic K709. 
