SHARYGIN 2014 Problem 20

A quadrilateral $KLMN$ is given. A circle with center $O$ meets its side $KL$ at points $A$ and $A_1$ , side $LM$ at points $B$ and $B_1$ , etc. Prove that if the circumcircles of triangles $KDA, LAB, MBC$ and $NCD$ concur at point $P$ , then a) the circumcircles of triangles $KD_1A_1, LA_1B_1, MB_1C_1$ and $NC1D1$ also concur at some point $Q$ ; b) point $O$ lies on the perpendicular bisector to $PQ$ .