We are given a triangle $\triangle ABC$ with the following properties:

• $AB = 18$, $BC = 24$, and $CA = 20$,
• Points $D$ and $E$ are placed such that $AD = 15$ (on $AB$) and $EC = 20$ (on $BC$),
• The intersection of lines $AE$ and $DC$ is denoted as $F$,
• We are asked to compute the expression: $\left( \text{Area of } \triangle DEF \right)^2 \cdot \frac{7^3 \cdot 6^3}{5^3}$

Step-by-Step Approach:

1. Determine the basic geometry of triangle ABC: We'll start by using Heron's formula or properties of triangles to determine the area of $\triangle ABC$. Once the area of the whole triangle is known, we can use the fact that $F$ is the intersection of the two lines to compute proportions of the sub-triangle $\triangle DEF$.

2. Utilize Mass Points or Menelaus’ Theorem: The geometry of the triangle suggests that mass point geometry or Menelaus' theorem can help in finding the ratios in which point $F$ divides the sides and how it affects the area of triangle $DEF$.

3. Apply the Area Ratios: Once we have the ratio for the area of triangle $DEF$ relative to $\triangle ABC$, we can square that area and multiply by the given expression $\frac{7^3 \cdot 6^3}{5^3}$.

I'll begin calculating now to get the exact values.The area of triangle $\triangle ABC$ is approximately $176.16$ square units.

Next, we need to calculate the area of triangle $\triangle DEF$, which involves determining how the point $F$ divides the triangle and the proportions of the area based on geometric principles like Menelaus' theorem or mass points.

Let me continue working through this calculation.To calculate the area of triangle $\triangle DEF$, we can use the following approach:

Ratios Involved

1. Mass Points or Menelaus Theorem:
   • Since $AD = 15$ and $AB = 18$, the ratio of $AD:DB = 15:3 = 5:1$.
   • For $E$, we are given $EC = 20$ and $BC = 24$, so $BE:EC = 4:20 = 1:5$.

Using these ratios, we can compute the area of triangle $\triangle DEF$ relative to $\triangle ABC$.

Step 2: Calculating the Area of Triangle DEF

By applying area ratios based on these segment divisions and properties of the intersection point $F$, we can find that the area of $\triangle DEF$ is a known fraction of the total area $\triangle ABC$. Let me compute the exact proportions and derive the area of triangle $DEF$.The area of triangle $\triangle DEF$ is approximately $4.89$ square units.

Step 3: Final Computation

We are asked to compute the following expression: $\left( \text{Area of } \triangle DEF \right)^2 \cdot \frac{7^3 \cdot 6^3}{5^3}$ Let's calculate this now.The final value of the expression is approximately $14,191.51$.

Would you like a detailed breakdown of any of the steps, or further explanation on a particular part?

Here are 5 related questions to extend your understanding:

1. How does Heron's formula work for calculating the area of any triangle?
2. What is Menelaus' theorem and how is it applied in geometric problems?
3. How can mass points geometry help solve complex geometric division problems?
4. How does the area ratio between sub-triangles relate to the overall triangle in such intersection problems?
5. How can similar geometric principles be applied to 3D geometry problems?

Tip: Understanding triangle area ratios can simplify many complex geometric problems, especially when dealing with intersecting lines and points inside triangles!