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X(39), X(512), X(1015), X(1500), X(2028), X(2029), X(14090)
A1, B1, C1 : extraversions of X(1015)
A2, B2, C2 : extraversions of X(1500)
U, V, W : vertices of the cevian triangle of X(39)
A', B', C' : vertices of the anticevian triangle of X(512)
other points below
The Steiner and Brocard inscribed ellipses meet at four points namely X(1015) and its three extraversions A1, B1, C1.
The triangle A1B1C1 is perspective
K554 is the pivotal cubic that contains all these points and also :
One of the asymptotes of K554 is the perpendicular at X(39) to the Brocard axis.
More generally, two inconics C(X), C(Y) with perspectors X, Y meet at four (all real or all imaginary) points lying on the pivotal cubic K(X, Y) whose pivot T is the crosspoint of X and Y, which is invariant under the isoconjugation with fixed point the intersection S of the trilinear polars of X and Y.
Recall that the trilinear polar L(S) of S is the fourth common tangent to C(X), C(Y).
K(X, Y) contains the vertices A'B'C' of the cevian triangle of T, the vertices QaQbQc of the anticevian triangle of S.