SHARYGIN 2013 Problem 97

Two fixed circles $\omega_1$ and $\omega_2$ pass through point $O$ . A circle of an arbitrary radius $R$ centered at $O$ meets $\omega_1$ at points $A$ and $B$ , and meets $\omega_2$ at points $C$ and $D$ . Let $X$ be the common point of lines $AC$ and $BD$ . Prove that all the points X are collinear as $R$ changes.