[ Measurement of segment lengths and angle measures. Adjacent angles.]

Point $B$ divides segment $A C$ in the ratio $A B: B C=2: 1$. Point $D$ divides segment $A B$ in the ratio $A D: D B=3: 2$. In what ratio does point $D$ divide segment $A C$?
Divide the segment $A C$ into 15 equal parts.

## Solution

We will divide the segment $A C$ into 15 equal parts. Then, the segment $B C$ will consist of 5 parts, the segment $A B$ will consist of 10 parts, the segment $A D$ will consist of 6 parts, the segment $B D$ will consist of 4 parts, and the segment $D C$ will consist of 4 + 5 = 9 parts. Therefore, $A D: D C = 6: 9 = 2: 3$.

## Answer

$A D: D C = 2: 3$.

Points $C, E$, and $D$ divide the segment $A B$ in the ratios $1: 2, 1: 3$, and $1: 4$ respectively (counting from point $A$).

In what ratio does point $E$ divide the segment $D C$?

## Hint

Divide the segment $A B$ into 60 equal parts.

## Solution

We will divide the segment $A B$ into 60 equal parts. Then, the segment $A C$ will consist of $60: 3 = 20$ parts, the segment $A E$ will consist of $60: 4 = 15$ parts, and the segment $A D$ will consist of $60: 5 = 12$ parts. Therefore, the segment $E C$ will consist of $20 - 15 = 5$ parts, and the segment $D E$ will consist of $15 - 12 = 3$ parts. Therefore, $D E: E C = 3: 5$.

## Answer

$D E: E C = 3: 5$.