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∑ SA x (SB y^2 - SC z^2) = 0

∑ SA x^2 (SB y - SC z) = 0

X(2), X(4), X(253), X(1249)

A', B', C' : midpoints of ABC

The polar conic of H is the Kiepert hyperbola and the tangents at A, B, C, G to K663 pass through H.

The polar conic of X(1249) – the tangential of H – is the complement of the rectangular circum-hyperbola through X(20). The tangents at A', B', C', H to K663 pass through X(1249).

K663 is the isogonal transform of pK(X184, X3), the isotomic transform of pK(X69, X69).

It meets the Steiner ellipse at the same points as K170 = pK(X2, X4).

Its asymptotes are parallel to those of pK(X2, X20) which is its anticomplement.

We meet this cubic in the paper "Another kind of Lemoine cubics".