BALKAN 1993 Problem 3

Circles $\mathcal C_1$ and $\mathcal C_2$ with centers $O_1$ and $O_2$ , respectively, are externally tangent at point $\lambda$ . A circle $\mathcal C$ with center $O$ touches $\mathcal C_1$ at $A$ and $\mathcal C_2$ at $B$ so that the centers $O_1$ , $O_2$ lie inside $C$ . The common tangent to $\mathcal C_1$ and $\mathcal C_2$ at $\lambda$ intersects the circle $\mathcal C$ at $K$ and $L$ . If $D$ is the midpoint of the segment $KL$ , show that $\angle O_1OO_2 = \angle ADB$ .

Greece