Let $\Omega_1(O_1, r_1)$ and $\Omega_2(O_2, r_2)$ be two circles that intersect in two points $X, Y$ . Let $A, C$ be the points in which the line $O_1O_2$ cuts the circle $\Omega_1$, and let $B$ be the point in which the circle $\Omega_2$ itnersect the interior of the segment $AC$, and let $M$ be the intersection of the lines $AX$ and $BY$ . Prove that $M$ is the midpoint of the segment $AX$ if and only if $O_1O_2 =\frac12 (r_1 + r_2)$.