Question 183
The circles $\omega_1$ and $\omega_2$ centered at $O_1$ and $O_2$, respectively, intersect at the point $X$. The line $O_1X$ intersects $\omega_2$ at the points $X$ and $B$, and the line $XO_2$ intersects $\omega_1$ at the points $X$ and $A$. The line $O_1A$ intersects $\omega_1$ at the points $A$ and $D$, and the line $O_2B$ intersects $\omega_2$ at the points $B$ and $C$. Let $M$ be the midpoint of the segment $CD$.
Prove that the lines $MX$ and $AB$ are perpendicular to each other.
Prove that the lines $MX$ and $AB$ are perpendicular to each other.