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X(2), X(15), X(16), X(512), X(3111)

K193 is the hessian cubic of Kjp = K024.

It is a focal cubic with focus G. It is the locus of contacts of tangents drawn from G to the circles of the pencil whose Poncelet (limit) points are the isodynamic points X(15) and X(16). Recall that when the circles pass through X(15) and X(16) we obtain K048 = McCay hessian cubic. See a generalization in this page.

The polar conic of G is the circle with diameter X(1340)X(1341), centers of similitude of the Brocard circle and the circumcircle. It is orthogonal to the Parry circle and the antipode of G in this circle is the point X where the curve meets its real asymptote, homothetic of the Lemoine axis under h(G,2).

K193 meets the Brocard axis at the isodynamic points X(15), X(16) and X(3111), point labelled Y on the figure, which lies on the parallel at G to the Lemoine axis.

The polar conic of the infinite point X(512) is the rectangular hyperbola passing through X(187), X(511), X(512). It meets K193 at four points which are the centers of anallagmaty of the curve. These points are the in/excenters of triangle GX(15)X(16). It follows that K193 is the isogonal pK with pivot X(512) with respect to this latter triangle.

K193 is also invariant under the Psi transformation which is the product of the reflection about one axis of the Steiner inellipse and the inversion with circle that of diameter F1F2, the foci of the ellipse. See "Orthocorrespondence and Orthopivotal Cubics", ยง5 and K018, K022, also K508, K816. Note that Psi swaps the two in/excenters above which lie on a same axis of the Steiner ellipse. K193 is a Psi-cubic as in Table 60.

Kjp and K193 obviously meet at the nine inflexion points (three real, six imaginary) of the two curves.