Question 179
The quadrilateral $ABCD$ is inscribed. The point $H_1$ is the orthocenter of $\triangle ABC$ and the points $A_1$ and $B_1$ are symmetric to the points $A$ and $B$ with respect to the lines $BH_1$ and $AH_1$, respectively. The point $O_1$ is the center of the circumscribed circle of $\triangle A_1B_1H_1$. The point $H_2$ is the orthocenter of $\triangle ABD$ and the points $A_2$ and $B_2$ are symmetric to the points $A$ and $B$ with respect to the lines $BH_2$ and $AH_2$, respectively. The point $O_2$ is the center of the circumscribed circle of $\triangle A_2B_2H_2$. Denote the line $O_1O_2$ by $l_{AB}$. The lines $l_{BC}$, $l_{CD}$, and $l_{DA}$ are defined analogously. Let $l_{AB} \cap l_{BC} = M$, $l_{BC} \cap l_{CD} = N$, $l_{CD} \cap l_{DA} = P$, and $l_{DA} \cap l_{AB} = Q$. Prove that the points $M, N, P$, and $Q$ are concyclic.