[ List pairs of similar triangles

Point $D$ lies on side $B C$ of triangle $A B C$, and point $O$ is located on segment $A D$, such that $A O: O D=9: 4$. The line passing through vertex $B$ and point $O$ intersects side $A C$ at point $E$, and $B O: O E=5: 6$. Find the ratio in which point $E$ divides side $A C$.

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Through vertex $B$, draw a line parallel to $A C$.

## Solution

Through point $B$, we will draw a line parallel to $A C$, and extend $A D$ to intersect this line at point $T$. From the similarity of triangles $B O T$ and $E O A$, it follows that $B T: A E = O T: A O = 5: 6$. Therefore, $T D = T O - D O = 5 / 6 A O - 4 / 9 A O = 7 / 18 A O = 7 / 26 A D$. From the similarity of triangles $B D T$ and $C D A$,

$A C = 26 / 7 B T = 26 / 7 \cdot 5 / 6 A E = 65 / 21 A E$. Therefore, $E C = A C - A E = 44 / 21 A E$.

![](https://cdn.mathpix.com/cropped/2024_05_06_f7f9d90261bfc416bd71g-18.jpg?height=352&width=552&top_left_y=1&top_left_x=753)

## Answer

$A E: E C = 21: 44$.