X(6), X(385), X(523), X(2574), X(2575), X(8105), X(8106), X(10221)
6 feet of the bisectors
vertices of the cevian triangle of X(648), the trilinear pole of the Euler line
Consider an isogonal pivotal cubic with pivot P on the Euler line and isopivot P* on the Jerabek hyperbola i.e. a member of the Euler pencil of cubics.
The poles of the line PP* in this cubic are the four common points of the polar conics of P and P*. Recall that the polar conic of P contains P and the four in/excenters, and that the polar conic of P* contains A, B, C, P, P*.
Hence one of these four points is P and there are three other common points, one at least being always real. When P traverses the Euler line, the locus of these three points is K511.
The figure shows the McCay cubic K003 and the two corresponding polar conics meeting at O, K, X(2574), X(2575).
K511 has three real asymptotes namely the lines X(110)X(2574), X(110)X(2575) and X(23)X(385).