51st Ukrainian National Mathematical Olympiad, 3rd Round

Two circles touch each other externally at point $C$. Consider two diameters $A_1{A}_2$, $B_1{B}_2$ of the same direction. Circle with the center on the common internal tangent passes through the point of intersection of $A_1{B}_2$, $A_2{B}_1$, and meets these lines at points $M$, $N$. Prove that $MN$ is perpendicular to $A_1{A}_2$, $B_1{B}_2$.