Let $k_{1}$ and $k_{2}$ be two circles with the same center, with $k_{2}$ inside $k_{1}$. Let $A$ be a point on $k_{1}$ and $B$ a point on $k_{2}$ such that $(A B)$ is tangent to $k_{2}$. Let $C$ be the second intersection of $(A B)$ and $k_{1}$ and let $D$ be the midpoint of $[A B]$. A line passing through $A$ intersects $k_{2}$ at $E$ and $F$ in such a way that the perpendicular bisectors of $[D E]$ and $[C F]$ intersect at a point $M$ such that $M$ lies on $(A B)$. Find the value of $\frac{A M}{M C}$.
On a $A E \cdot A F=A B^{2}=4 A D^{2}=A D \cdot A C$ so $E, F, C$ and $D$ are concyclic. The perpendicular bisectors of $[B E]$ and $[C F]$ intersect on $(C D)$ so $M$ is the center of the circumcircle of $E F C D$. We deduce $\frac{A M}{M C}=\frac{A C-M C}{M C}=\frac{2 \cdot A C}{D C}-1=\frac{8}{3}-1=\frac{5}{3}$.