11. (IND $\left.3^{\prime}\right)^{\mathrm{IMO}}$ Given a circle with two chords $A B, C D$ that meet at $E$, let $M$ be a point of chord $A B$ other than $E$. Draw the circle through $D, E$. and $M$. The tangent line to the circle $D E M$ at $E$ meets the lines $B C, A C$ at $F, G$, respectively: Given $\frac{A M}{A G}=\lambda$, find $\frac{G E}{E Y}$.
11. Assume $B(A, E, M, B)$. Since $A, B, C, D$ lie on a circle, we have $\angle G C E=$ $\angle M B D$ and $\angle M A D=\angle F C E$. Since $F D$ is tangent to the circle around $\triangle E M D$ at $E$, we have $\angle M D E=\angle F E B=\angle A E G$. Consequently, $\angle C E F=180^{\circ}-\angle C E A-\angle F E B=180^{\circ}-\angle M E D-\angle M D E=\angle E M D$ and $\angle C E G=180^{\circ}-\angle C E F=180^{\circ}-\angle E M D=\angle D M B$. It follows that $\triangle C E F \sim \triangle A M D$ and $\triangle C E G \sim$ $\triangle B M D$. From the first similarity we obtain $C E \cdot M D=A M \cdot E F$, and from the second we obtain $C E$. $M D=B M \cdot E G$. Hence
$$
\begin{array}{c}
A M \cdot E F=B M \cdot E G \Longrightarrow \\
\frac{G E}{E F}=\frac{A M}{B M}=\frac{\lambda}{1-\lambda} .
\end{array}
$$

If $\mathcal{B}(A, M, E, B)$, interchanging the roles of $A$ and $B$ we similarly obtain $\frac{G E}{E F}=\frac{\lambda}{1-\lambda}$.