Školjka

Two circles with centers $O_{1}$ and $O_{2}$ meet at two points $A$ and $B$ such that the centers of the circles are on opposite sides of the line $\overline{AB}$. The lines $BO_{1}$ and $BO_{2}$ meet their respective circles again at $B_{1}$ and $B_{2}$. Let $M$ be the midpoint of $\overline{B_{1}B_{2}}$. Let $M_{1}$, $M_{2}$ be points on the circles of centers $O_{1}$ and $O_{2}$ respectively, such that $\angle AO_{1}M_{1}= \angle AO_{2}M_{2}$, and $B_{1}$ lies on the minor arc $\overset{\frown}{AM_{1}}$ while $B$ lies on the minor arc $\overset{\frown}{AM_{2}}$. Show that $\angle MM_{1}B = \angle MM_{2}B$.