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K596

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X(74), X(99), X(265), X(290)

K596 is a nodal stelloid with node the Steiner point X(99) which is a double pivot. The simple pivot is X(265). Hence, any circle passing through X(99) and X(265) meets the cubic again at three points which are the vertices of an equilateral triangle.

It has three real asymptotes concurring at X = X(14850). See K595 for a construction of these asymptotes.

K595 and K596 generate a pencil of circum-stelloids that contains the cubic decomposed into the line at infinity and the Jerabek hyperbola. It follows that K595 and K596 have the same points at infinity (hence parallel asymptotes) and meet at six finite points on the Jerabek hyperbola which are A, B, C, X(74), X(265), X(290).

K596a

Let (K) be a cubic of the pencil.

(K) meets the circumcircle (O) at A, B, C, X(74) and two (not always real) points O1, O2 on a line passing through X(804) i.e. parallel to the tangents at X(98), X(99) to (O).

The third point O3 of (K) on this line also lies on the line passing through X(265), X(290).

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(K) meets the Steiner ellipse (S) at A, B, C, X(290) and two (not always real) points S1, S2 on a line passing through X(690) i.e. parallel to the tangents at X(99), X(671) to (S).

The third point S3 of (K) on this line also lies on the line passing through X(30), X(74), X(265), X(476).

***

The radial center X of (K) lies on a parallel to the line X(98)X(99) passing through the radial centers X(14849), X(14850) of K595, K596 respectively.