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K615

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X(2), X(3), X(4), X(64), X(154), X(3424), X(5373)

infinite points of the altitudes of ABC

points of the Thomson cubic K002 on the circumcircle i.e. vertices of the Thomson triangle

excenters of the Thomson triangle, the incenter being X(5373). Note that their reflections about O lie on the Stammler hyperbola

K615 is the Thomson cubic's sister. See the related K376, K405 and K763 and other analogous cubics in Table 58.

K615 is the isogonal transform of the Antreas cubic K047.

It is the pivotal isogonal cubic with pivot H with respect to the Thomson triangle T. It is therefore invariant under isogonal conjugation with respect to T. Hence, it is a member of the Euler pencil of cubics in T also containing K078, the McCay cubic of T, and K764, the Darboux cubic of T.

The tangents to K615 at the vertices of the Thomson triangle concur at X(154).

K615 belongs to the pencils generated :

  • by the Darboux cubic K004 and the union of the Jerabek hyperbola with the line at infinity,
  • by the Thomson cubic K002 and the union of the circumcircle with the Euler line. See also K581.

K615 is spK(X20, X376) in CL055. See also K759.