Question 141
Ordered four points $A$, $B$, $C$, $D$ lie on a given circle. Let $AC$ and $BD$ segments meet at $M$. A line passing through the point $M$ and given circle meets at $M_1$ and $M_2$, which line and the circles $\omega(ABM)$ and $\omega(CDM)$ meets at $N_1$ and $N_2$ respectively. Show that $M_1N_1 = M_2N_2$.
(proposed by B. Ganbileg and U. Batzorig)
(proposed by B. Ganbileg and U. Batzorig)