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.