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X(4), X(278), X(1249)

foci of the circum-conic (C) with perspector X(4), center X(1249)

infinite points of pK(X6, X253)

Ha, Hb, Hc :vertices of the orthic triangle

A', B', C' : points of (C) on K663

Consider a circum-conic (C) with perspector P. The circum-cubics passing through its four foci form a net of cubics which contain K0s when P lies on the Kiepert hyperbola and psKs when P lies on the Thomson cubic K002.

It follows that there are only two points P such that the net contains a pK. These are P = G giving the Lucas cubic K007 passing through the foci of the Steiner circum-ellipse and P = H giving the cubic K709.

Note that the pole X(6525) of K709 is the barycentric product of X(4) and X(1249). In other words, K709 is invariant under the isoconjugation which swaps the perspector and the center of (C).