In rectangle $A B C D$, points $E$ and $F$ are chosen on sides $A B$ and $A D$ respectively such that $A E: E B=3: 1$, $A F$ $: F D=1: 2$.

Find the ratio $E O: O D$, where $O$ is the point of intersection of segments $D E$ and $C F$.

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Continue segments $A B$ and $C F$ until they intersect at point $M$ and consider two pairs of similar triangles: $A F M$, $D F C$ and $M O E, C O D$.

## Solution

$C D=A B=4 B E$. Let the lines $A B$ and $C F$ intersect at point $M$. From the similarity of triangles $A F M$ and $D F C$, it follows that $A M=1 / 2 C D=2 B E$.

From the similarity of triangles $M O E$ and $C O D$, it follows that $E O: O D=M E: C D=5: 4$.

![](https://cdn.mathpix.com/cropped/2024_05_06_4c124ad32a76c9faf91dg-40.jpg?height=501&width=501&top_left_y=-5&top_left_x=784)

## Answer

$5: 4$.

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