An elimination problem

Given a triangle $ABC$ and two isogonal points $P$ and $P*$, call $A'B'C'$ and $A''B''C''$ the antipedal triangles of $P, P*$, respectively.

The triangle bounded by $A'A''$, $B'B''$, $C'C''$ is always perspective with ABC (see Hyacinthos message #21782).

The perspector is complicated, although we can see that it is the isotomic conjugate of a simpler point.
We want to calculate the locus of this isotomic conjugate when the point $P$ moves along the Euler line. 


An elimination problem