MathNet 2015-part-03 — 191 problems (EN–VI) Question 7

Let $ABCD$ be a convex quadrilateral (that is, its diagonals lie inside $ABCD$), and let $P$ be the intersection of its diagonals $[AC]$ and $[BD]$. Let $O_1, O_2, O_3$ and $O_4$ be the centers of the circumcircles of $ABP$, $BCP$, $CDP$ and $DAP$.
Show that $O_1O_2O_3O_4$ is a parallelogram.