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X(3413) on K400a

X(3414) on K400b

The Steiner central cubics are the nK0++ whose pole is the infinite point of one of the axes of the Steiner ellipse and whose root is the infinite point of the other axis.

The pole of K400a and the root of K400b is the infinite point X(3413) of the focal axis (passing through the real foci).

These two cubics meet at A, B, C, their reflections A', B', C' in G and G with a common inflexional tangent which is the line GK.

The asymptotes of K400a are the focal axis of the Steiner ellipse and the parallel at G to the asymptotes of the circum-conic with centre the reflection in G of the barycentric square of its root. Those of K400b similarly. These centres are the blue points on the figure. They obviously lie on the Steiner inellipse. The tangents at these points are the trilinear polars of the roots of the cubics.

Each cubic meets the asymptotes of the other at six points lying on the ellipse homothetic of the Steiner inellipse under h(G, √2). These are the dark green points on the figure.