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X(351), X(512), X(804), X(878), X(881), X(9178)

A', B', C' : traces of the trilinear polar of X(3124)

Geometric properties :

K978 is the isogonal transform of K052. See here for a family of related cubics.

It is tritangent at A, B, C to the circumcircle.

It has three real asymptotes :

• one is the line X(76)X(804), the perpendicular at X(76) to the line X(3)X(76),

• two are the lines X(512)X(2028), X(512)X(2029) hence parallel and perpendicular to the Brocard axis.

The union of these three asymptotes is symmetric about S = barycentric product X(512) x X(13518) = barycentric quotient X(13518) ÷ X(670). S lies on the lines {3,887}, {39,512}, {76,804}, {669,11182}, {4173,9429}.

The pivotal conic (C) is inscribed in the anticevian triangle of X(512) and it is tangent to the parallel asymptotes. (C) also contains the root X(3124).