It is known that circle $\Gamma_1(O_1)$ has center at $O_1$, circle $\Gamma_2(O_2)$ has center at $O_2$, and both intersect at points $C$ and $D$. It is also known that points $P$ and $Q$ lie on circles $\Gamma_1(O_1)$ and $\Gamma_2(O_2)$, respectively. ). A line $\ell$ passes through point $D$ and intersects $\Gamma_1(O_1)$ and $\Gamma_2(O_2)$ at points $A$ and $B$, respectively. The lines $PD$ and $AC$ meet at point $M$, and the lines $QD$ and $BC$ meet at point $N$. Let $O$ be center outer circle of triangle $ABC$. Prove that $OD$ is perpendicular to $MN$ if and only if a circle can be found which passes through the points $P, Q, M$ and $N$.
Problem
Source: Indonesia INAMO Shortlist 2017 G10 https://artofproblemsolving.com/community/c1101409_indonesia_shortlist__geometry
Tags: perpendicular, Concyclic, geometry