[ Two pairs of similar triangles ]

Points $K$ and $M$ lie on sides $A B$ and $B C$ of triangle $A B C$, such that $A K: B K=3: 2, B M: M C=3: 1$. A line $l$ is drawn through point $B$, parallel to $A C$. Line $K M$ intersects line $l$ at point $P$, and line $A C$ at point $N$. Find $B P$ and $C N$, if $A C=a$.
Consider two pairs of similar triangles.

## Solution

From the similarity of triangles $C M N$ and $B M P$, it follows that $B P=3 C N$, and from the similarity of triangles $A K N$ and $B K P$, we have $A N = 3 / 2 B P = 9 / 2 C N$.

Thus, $a=A C=7 / 2 C N$. Therefore, $C N=2 a / 7$ and $B P=6 a / 7$.

![](https://cdn.mathpix.com/cropped/2024_05_06_96d74c2697cd905bd948g-13.jpg?height=317&width=640&top_left_y=2294&top_left_x=709)

## Answer

$6 a / 7, 2 a / 7$.

Given triangle $A B C$. On the extension of side $A C$ beyond point $C$, point $N$ is taken such that $A C=2 C N$. Point $M$ is on side $B C$, and $B M: M C=1: 3$.

In what ratio does the line $M N$ divide side $A B$?

## Solution

First method. Draw a line through point $B$ parallel to $A C$. Let the line $M N$ intersect it at point $T$, and the line $A B$ at point $K$.

$C N=1 / 2 A C, A N=3 / 2 A C$. From the similarity of triangles $T B M$ and $N C M$ (coefficient $1 / 3$), we find that $T B=1 / 3 C N = 1 / 6 A C$, and from the similarity of triangles $T B K$ and $N A K$, we have $B K: A K = B T: A N = 1: 9$.

![](https://cdn.mathpix.com/cropped/2024_05_06_96d74c2697cd905bd948g-14.jpg?height=588&width=600&top_left_y=615&top_left_x=732)

Second method. Draw a line through point $M$ parallel to $A B$ until it intersects $A C$ at point $L$. From the similarity of triangles $L M C$ and $A B C$, we find that $L M=3 / 4 A B$ and $A L=1 / 4 B C=1 / 6 A N$, and from the similarity of triangles $L M N$ and $A K N$, we have $A K=6 / 5 \cdot 3 / 4 A B=9 / 10 A B$.

## Answer

$1: 9$, measured from point $B$.