BALKAN 1995 Problem 2

The circles $\mathcal C_1(O_1, r_1)$ and $\mathcal C_2(O_2, r_2)$ , $r_2 > r_1$ , intersect at $A$ and $B$ such that $\angle O_1AO_2 = 90^\circ$ . The line $O_1O_2$ meets $\mathcal C_1$ at $C$ and $D$ , and $\mathcal C_2$ at $E$ and $F$ (in the order $C$ , $E$ , $D$ , $F$ ). The line $BE$ meets $\mathcal C_1$ at $K$ and $AC$ at $M$ , and the line $BD$ meets $\mathcal C_2$ at $L$ and $AF$ at $N$ . Prove that $$\frac{ r_2}{r_1} = \frac{KE}{KM} \cdot \frac{LN}{LD} .$$ Greece