BALKAN 1997 Problem 3

The circles $\mathcal C_1$ and $\mathcal C_2$ touch each other externally at $D$ , and touch a circle $\omega$ internally at $B$ and $C$ , respectively. Let $A$ be an intersection point of $\omega$ and the common tangent to $\mathcal C_1$ and $\mathcal C_2$ at $D$ . Lines $AB$ and $AC$ meet $\mathcal C_1$ and $\mathcal C_2$ again at $K$ and $L$ , respectively, and the line $BC$ meets $\mathcal C_1$ again at $M$ and $\mathcal C_2$ again at $N$ . Prove that the lines $AD$ , $KM$ , $LN$ are concurrent.

Greece