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foci of the Steiner circum-ellipse and their reflections in H

The Lucas cubic K007 and the Ehrmann strophoid K025 generate a pencil of circum-cubics passing through H and the (four) foci of the Steiner circum-ellipse. All these cubics share the same tangent at H (the line HX(51)) and the polar conic at H is always a rectangular hyperbola centered on the line HX(74).

This pencil contains one equilateral cubic K60+ which is K309 and one central cubic K++ which is K310.

K310 has three (not always real) asymptotes concurring at the center H and the inflexional tangent at H passes through X(51).