X(3), X(4), X(376), X(1340), X(1341)
foci of the Steiner circum-ellipse
points at infinity of the McCay cubic
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 K++ which is K310.
K309 has three real asymptotes concurring at X = X(5054) such that OX = 2/9 OH and these asymptotes are parallel to those of the McCay cubic.
K309 is spK(X3, X8703) as in CL055.
K309 meets the circumcircle at the same points as K243 = pK(X6, X376).