8,9,10 |
| :---: | :---: | :---: |
|  | Similarity Criteria |  |
|  | Thales' Theorem and the Proportional Segments Theorem |  |
|  | Ceva's Theorem and Menelaus' Theorem |  |

On the sides $AB$ and $BC$ of triangle $ABC$, points $M$ and $N$ are located respectively, such that $AM: MB = 3: 5, BN: NC = 1: 4$. Lines $CM$ and $AN$ intersect at point $O$. Find the ratios $OA: ON$ and $OM: OC$.
Draw a line through vertex $A$ parallel to side $B C$, and consider two pairs of similar triangles.

## Solution

Draw a line through point $A$ parallel to $B C$. Let $T$ be the point of intersection of this line with the line $M C$. Let $B N=a, C N=4 a$.

From the similarity of triangles $A M T$ and $B M C$ (with a similarity coefficient of $3 / 5$), we find that $A T=3 / 5 B C=3 / 5(B N+N C)=3 / 5(a+4 a)$ $=3 a$, and from the similarity of triangles $A O T$ and $N O C$, we have $O A: O N=A T: C N=3: 4$.

Similarly, we find that $O M: O C=3: 32$.

## Answer

$3: 4 ; 3: 32$.

![](https://cdn.mathpix.com/cropped/2024_05_06_a69fc833ac69b1da7555g-07.jpg?height=283&width=2035&top_left_y=99&top_left_x=0)

On the sides $A B$ and $A C$ of triangle $A B C$, points $N$ and $M$ are located, respectively, such that $A N: N B=3: 2, A M: M C=4: 5$. Lines $B M$ and $C N$ intersect at point $O$. Find the ratios $O M: O B$ and $O N: O C$.

## Solution

See the solution to problem $\underline{116338}$.

## Answer

$5: 6 ; 8: 25$