Question 135
The diagonals $AC$ and $BD$ of the convex quadrilateral $ABCD$ intersect at point $O$. The points $A_1$, $B_1$, $C_1$ and $D_1$ on the segments $AO$, $BO$, $CO$ and $DO$, respectively, are such that $AA_1 = CC_1$ and $BB_1 = DD_1$. Let $M$ be the second intersection point of the circumcircles of $\triangle AOB$ and $\triangle COD$; $N$ be the second intersection point of the circumcircles of $\triangle AOD$ and $\triangle BOC$; $P$ be the second intersection point of the circumcircles of $\triangle A_1OB_1$ and $\triangle C_1OD_1$, and $Q$ be the second intersection point of the circumcircles of $\triangle A_1OD_1$ and $\triangle B_1OC_1$. Prove that the points $M$, $N$, $P$ and $Q$ are concyclic.