KöMaL 2019 B5060

In the plane $\Sigma$, given a circle $k$ and a point $P$ in its interior, not coinciding with the center of $k$. Call a point $O$ of space, not lying on $\Sigma$, a proper projection center if there exists a plane $\Sigma'$, not passing through $O$, such that, by projecting the points of $\Sigma$ from $O$ to $\Sigma'$, the projection of $k$ is also a circle, and its center is the projection of $P$. Show that the proper projection centers lie on a circle.