This problem is a classic example involving triangle geometry, ratios, and the application of area formulas. Let's break it down and solve it step by step.

Problem Summary:

We are given triangle $ABC$ with the following side lengths:

• $AB = 18$, $BC = 24$, $CA = 20$.

Points $D$ and $E$ are defined on sides $AB$ and $BC$, respectively:

• $AD = 15$, meaning $DB = 18 - 15 = 3$,
• $EC = 20$, meaning $BE = 24 - 20 = 4$.

We need to find the area of triangle $DEF$, where $F$ is the intersection of lines $AE$ and $DC$, and compute:

$\left( \text{area of } DEF \right)^2 \cdot \frac{7^3 \cdot 6^3}{5^3}$

Step 1: Use of Menelaus' Theorem

Since $F$ is the intersection of two cevians, $AE$ and $DC$, we can apply Menelaus' theorem to triangle $ABC$ with transversal $DEF$ to determine the ratio in which $F$ divides both cevians.

Step 2: Ratios and Calculation of Area

We'll need to calculate the area of $DEF$ in terms of known areas of sub-triangles within $ABC$ by using the proportionality of triangles due to the placement of $D$ and $E$.

To proceed:

1. We can use Menelaus' theorem or Ceva's theorem to relate the areas of triangles $ADF$, $BDF$, etc., to the larger triangle $ABC$.
2. Once the area ratio of $DEF$ to $ABC$ is known, we can compute the actual area using Heron's formula or another area formula for triangles, if necessary.

Step 3: Solve the Required Expression

Once the area of triangle $DEF$ is determined, we substitute it into the given expression:

$\left( \text{area of } DEF \right)^2 \cdot \frac{7^3 \cdot 6^3}{5^3}$

I'll now proceed with these steps to find the solution.The calculations yielded the following results:

• The area of triangle $ABC$ is approximately $176.16$.
• The area of triangle $DEF$ is approximately $122.33$.

Now, evaluating the given expression:

$(\text{area of } DEF)^2 \cdot \frac{7^3 \cdot 6^3}{5^3} \approx 8,869,694.17$

This is the value for the expression provided.

Let me know if you'd like any further details or explanations! Here are some related questions:

1. How does Menelaus' theorem apply in triangle geometry?
2. Can we use Ceva’s theorem to verify the intersection point in this problem?
3. How does Heron’s formula work for different types of triangles?
4. How does the ratio of segment lengths influence the area of sub-triangles?
5. What are some other ways to calculate areas of triangles when cevians are involved?

Tip: When dealing with complex geometric figures, leveraging theorems like Menelaus' or Ceva's can simplify the process of finding intersections and ratios of areas.