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This problem is asking for the ratio of the areas of two triangles within the larger triangle ABC.

### Given Information:
- \( \triangle ABC \) is a triangle.
- Points \( E \) and \( D \) are placed on sides \( AB \) and \( AC \), respectively.
- The following relations are provided:
  - \( |AE| = |EB| \)
  - \( |DC| = |AD| \)

### Objective:
To find the ratio:
\[
\frac{\text{Area}(\triangle ABC)}{\text{Area}(\triangle AED)}
\]

### Approach:

1. **Identify the Triangles**:
   - \( \triangle ABC \) is the entire triangle.
   - \( \triangle AED \) is a smaller triangle formed within \( \triangle ABC \).

2. **Apply Proportions Based on Given Lengths**:
   - From \( |AE| = |EB| \), point \( E \) divides side \( AB \) in a ratio of 1:1, meaning \( E \) is the midpoint of \( AB \).
   - Similarly, \( |AD| = |DC| \) indicates that \( D \) is the midpoint of \( AC \).

3. **Conclusion from Midpoints**:
   - Since both \( E \) and \( D \) are midpoints, \( \triangle AED \) is a smaller triangle inside \( \triangle ABC \) formed by connecting midpoints.
   - This configuration is known to create a triangle whose area is exactly one-quarter of the original triangle's area (since connecting midpoints in a triangle divides it into four smaller triangles of equal area).

4. **Area Ratio**:
   \[
   \frac{\text{Area}(\triangle ABC)}{\text{Area}(\triangle AED)} = 4
   \]

Thus, the answer is **C) 4**.

Would you like more details or have any questions?

### Related Questions:
1. How do we calculate the area of triangles using the base and height formula?
2. What other ways can midpoints divide a triangle, and how do they affect area?
3. How does the concept of similarity apply to triangles with proportional sides?
4. Can this problem be extended to 3D geometry, and how would the approach change?
5. How can coordinate geometry help in solving problems involving triangle area ratios?

**Tip**: In triangle geometry, dividing by midpoints often results in simple area ratios—use this to your advantage when solving similar problems.

Buna göre, Alan(ABC) / Alan(AED) oranı kaçtır?

Learn how to calculate the area ratio between triangles ABC and AED using midpoint theorems. This geometric problem demonstrates how dividing a triangle by midpoints affects area ratios. Suitable for students in Grades 8-10.

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