[ Two pairs of similar triangles ]

In parallelogram $A B C D$, points $E$ and $F$ lie on sides $A B$ and $B C$, respectively, and $M$ is the intersection point of lines $A F$ and $D E$, with $A E=2 B E$,

and $B F=3 C F$. Find the ratio $A M: M F$.
Let lines $B C$ and $D E$ intersect at point $K$. Triangle $K B E$ is similar to triangle $D A E$, and triangle $A M D$ is similar to triangle $F M K$.

## Solution

Let $K$ be the point of intersection of lines $B C$ and $D E$. Denote $A D=4 a$. Then $B F=3 / 4 B C=3 / 4 A D=3 a$.

From the similarity of triangles $K B E$ and $D A E$, we find that $B K=A D^{B E}{ }_{A E}=2 a$, and from the similarity of triangles $A M D$ and

$F M K-A M: M F=A D: F K=4: 5$.

## Answer

## $4: 5$.