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K616

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X(4), X(69), X(376), X(1249), X(5485), X(6601)

isogonal conjugate of X(3167)

infinite points of the Thomson cubic.

points of the Steiner ellipse on K170

points of (O) on pK(X6, Y) where Y = X(15305) is the reflection of X(3060) about X(4)

K616 is a nodal cubic with node H we meet in the page K609. See a generalization and other related cubics in Table 43 where it is the cubic (K) obtained when P = X(376). See also CL069 for similar cubics.

The nodal tangents are parallel to the asymptotes of the Jerabek hyperbola and to the axes of the Steiner ellipse.

The asymptotes of K616 are parallel to those of the Thomson cubic K002.

The isogonal transform of K616 is K626.

Constructions

Let S be a variable point on the circumcircle. The Steiner line (S) of S meets the line S-X(376) at Q. When S traverses the circumcircle, the locus of Q is the nodal cubic K616.

A variable line (L) passing through H meets the line passing through X(69) and the isotomic conjugate of the infinite point of (L) at a point on K616.