The crosspoint

We study some loci related to the construction known as crosspoint of two given points. The main result is as follows:

Let $ABC$ a triangle, $P$ a point and $L$ the trilinear polar of some point $M$. Then the locus of points $Q$ for which the crosspoint of $P$ and $Q$ lies on $L$ is the conic with perspector the crosspoint of  $P$ and $M$.

See details at The crosspoint