Question 1
The circles $\omega_1$ and $\omega_2$ centered at $O_1$ and $O_2$, respectively, intersect at the point $X$. The line $XO_1$ intersects $\omega_1$ at the points $X$ and $A$. The line $XO_2$ intersects $\omega_2$ at the points $X$ and $B$. Let $M$ be the midpoint of the segment $AB$. The ray $MO_1$ intersects $\omega_1$ at the point $D$, and the ray $MO_2$ intersects $\omega_2$ at the point $C$.
a) Prove that the triangle $MDC$ is isosceles.
b) Prove that the line $DC$ passes through $X$.
a) Prove that the triangle $MDC$ is isosceles.
b) Prove that the line $DC$ passes through $X$.