BG-MO 2018 Problem 2

Let $ABCD$ be a cyclic quadrilateral. Let $H_{1}$ be the orthocentre of triangle $ABC$ . Point $A_{1}$ is the image of $A$ after reflection about $BH_{1}$ . Point $B_{1}$ is the image of of $B$ after reflection about $AH_{1}$ . Let $O_{1}$ be the circumcentre of $(A_{1}B_{1}H_{1})$ . Let $H_{2}$ be the orthocentre of triangle $ABD$ . Point $A_{2}$ is the image of $A$ after reflection about $BH_{2}$ . Point $B_{2}$ is the image of of $B$ after reflection about $AH_{2}$ . Let $O_{2}$ be the circumcentre of $(A_{2}B_{2}H_{2})$ . Lets denote by $\ell_{AB}$ be the line through $O_{1}$ and $O_{2}$ . $\ell_{AD}$ , $\ell_{BC}$ , $\ell_{CD}$ are defined analogously. Let $M=\ell_{AB} \cap \ell_{BC}$ , $N=\ell_{BC} \cap \ell_{CD}$ , $P=\ell_{CD} \cap \ell_{AD}$ , $Q=\ell_{AD} \cap \ell_{AB}$ . Prove that $MNPQ$ is cyclic.