Consider triangle ABC, points D, E, and F are located on sides AB, BC, and AC respectively. Let BD:DA = 2:3, BE:EC = 2:3, and CF:FA = 2:3. Connect D and F, D and E, and E and F. If the area of triangle ADF is 18, what's the area of triangle DEF?

Let's consider triangle \(ABC\) and the points \(D\), \(E\), and \(F\) on sides \(AB\), \(BC\), and \(AC\) respectively. The key information is that the ratios of the line segments on each side are given as follows:

- \(BD : DA = 2 : 3\)
- \(BE : EC = 2 : 3\)
- \(CF : FA = 2 : 3\)

We are tasked with finding the area of triangle \(DEF\), given that the area of triangle \(ADF\) is 18.

### Step 1: Express the Ratios
From the problem, we have the following segment ratios:
- On side \(AB\), \(BD : DA = 2 : 3\). This implies that \(BD = \frac{2}{5} AB\) and \(DA = \frac{3}{5} AB\).
- On side \(BC\), \(BE : EC = 2 : 3\). This implies that \(BE = \frac{2}{5} BC\) and \(EC = \frac{3}{5} BC\).
- On side \(AC\), \(CF : FA = 2 : 3\). This implies that \(CF = \frac{2}{5} AC\) and \(FA = \frac{3}{5} AC\).

### Step 2: Analyze the Areas of Triangles
Let's assign the area of the entire triangle \(ABC\) as \(S\). The goal is to find the area of triangle \(DEF\) in terms of \(S\), and then use the given area of triangle \(ADF = 18\) to solve for the area of triangle \(DEF\).

#### Area of Triangle \(ADF\)
From the segment ratios on sides \(AB\) and \(AC\), we know:
- \(DA = \frac{3}{5} AB\)
- \(FA = \frac{3}{5} AC\)

Therefore, triangle \(ADF\) is similar to triangle \(ABC\), and the ratio of their areas is given by the product of the ratios of the sides:
\[
\text{Area of } ADF = \left(\frac{3}{5}\right)^2 \times S = \frac{9}{25} S
\]
Given that the area of triangle \(ADF = 18\), we have:
\[
\frac{9}{25} S = 18
\]
Solving for \(S\), the area of triangle \(ABC\), we get:
\[
S = \frac{18 \times 25}{9} = 50
\]
So, the area of triangle \(ABC\) is \(S = 50\).

### Step 3: Area of Triangle \(DEF\)
Now, we need to find the area of triangle \(DEF\). The points \(D\), \(E\), and \(F\) divide each side of triangle \(ABC\) in the ratio \(2:3\). This configuration suggests that triangle \(DEF\) is similar to triangle \(ABC\), and the ratio of their areas depends on the product of the division ratios.

The ratio of the area of triangle \(DEF\) to the area of triangle \(ABC\) is the product of the division ratios on each side:
\[
\text{Area ratio of DEF to ABC} = \left(\frac{2}{5}\right)^2 = \frac{4}{25}
\]
Thus, the area of triangle \(DEF\) is:
\[
\text{Area of } DEF = \frac{4}{25} \times 50 = 8
\]

### Final Answer:
The area of triangle \(DEF\) is \( \boxed{8} \).

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Would you like more details on the geometric reasoning or any clarification? 

Here are some related questions for further exploration:
1. What is the general formula for the area of a triangle given side ratios?
2. How does the division of sides affect the similarity of triangles?
3. Can this method be generalized for any ratio \(m:n\)?
4. What is the relationship between the centroid and the area of sub-triangles?
5. How do different placements of points \(D\), \(E\), and \(F\) affect the area of triangle \(DEF\)?

**Tip**: Understanding area ratios in triangles involves not just the linear division of sides but also the quadratic relationship in their areas.

Solve the problem of finding the area of triangle DEF in a triangle ABC, where points D, E, and F divide sides in the ratio 2:3. Learn the concept of area ratios in similar triangles and apply key theorems like the Area of Similar Triangles Theorem. Suitable for high school geometry students.

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