For a general triangle $ABC$, points $D, E, F$ are given as follows:

- point $D$ is one-third of the way along segment $AB$, closer to point $A$,
- point $E$ is one-third of the way along segment $BC$, closer to point $B$,
- point $F$ is one-third of the way along segment $CD$, closer to point $C$.

Determine the ratio of the areas of triangles $ABC$ and $DEF$.

(A. Bohiniková)
For clarity, we first illustrate the situation:

![](https://cdn.mathpix.com/cropped/2024_04_17_2593db36155ed0011264g-3.jpg?height=574&width=876&top_left_y=1415&top_left_x=590)

Triangles $A B C$ and $D B C$ have the same height from vertex $C$, and the opposite sides $(A B$ and $D B)$ are in the ratio $3: 2$. Consequently, their areas are also in the same ratio,

$$
S_{A B C}: S_{D B C}=3: 2 .
$$

Triangles $B C D$ and $E C D$ have the same height from vertex $D$, and the opposite sides $(B C$ and $E C)$ are in the ratio $3: 2$. Consequently, their areas are also in the same ratio, which, together with the previous result, gives

$$
S_{A B C}: S_{E C D}=9: 4
$$

Triangles $C D E$ and $F D E$ have the same height from vertex $E$, and the opposite sides $(C D$ and $F D)$ are in the ratio $3: 2$. Consequently, their areas are also in the same ratio, which, together with the previous results, gives

$$
S_{A B C}: S_{D E F}=27: 8
$$

Evaluation. 1 point for illustrating the situation; 1 point for each of the partial observations; 2 points for the conclusion and quality of the commentary.