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X(4) foci of the Steiner circumellipse and their reflections in H 

The Lucas cubic K007 and the Ehrmann strophoid K025 generate a pencil of circumcubics passing through H and the (four) foci of the Steiner circumellipse. 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). 
