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To solve the given problem step by step, let's break it down and solve each part systematically.

### Problem Statement
In \(\triangle ABC\), segment \(DE \parallel BC\). If \(2A(\triangle ADE) = A(\triangle DBCE)\), find \(AB : AD\) and show that \(BC = \sqrt{3} \cdot DE\).

### Solution

#### Step 1: Understanding the Relationship Between the Areas
Given:
\[ 2A(\triangle ADE) = A(\triangle DBCE) \]

We know that \(DE \parallel BC\), hence \(\triangle ADE \sim \triangle ABC\) by AA similarity criterion.

#### Step 2: Using the Area Proportionality
Since \(DE \parallel BC\), the triangles \(\triangle ADE\) and \(\triangle ABC\) are similar. Therefore, the ratio of their areas is the square of the ratio of their corresponding sides:
\[ \frac{A(\triangle ADE)}{A(\triangle ABC)} = \left(\frac{AD}{AB}\right)^2 \]

From the given, we have:
\[ 2A(\triangle ADE) = A(\triangle DBCE) \]
Since \(DE \parallel BC\), \( \triangle DBCE \) is composed of \(\triangle ADE\) and the parallelogram \(DEBC\). Therefore:
\[ A(\triangle DBCE) = A(\triangle ABC) - A(\triangle ADE) \]

Substituting this into the equation:
\[ 2A(\triangle ADE) = A(\triangle ABC) - A(\triangle ADE) \]
\[ 3A(\triangle ADE) = A(\triangle ABC) \]

Thus:
\[ \frac{A(\triangle ADE)}{A(\triangle ABC)} = \frac{1}{3} \]

From the similarity of triangles:
\[ \left(\frac{AD}{AB}\right)^2 = \frac{1}{3} \]

Taking the square root of both sides:
\[ \frac{AD}{AB} = \frac{1}{\sqrt{3}} \]
\[ AB = \sqrt{3} \cdot AD \]

Therefore:
\[ \frac{AB}{AD} = \sqrt{3} \]

#### Step 3: Showing that \(BC = \sqrt{3} \cdot DE\)
Since \( \triangle ADE \sim \triangle ABC \), the ratio of the sides is:
\[ \frac{AD}{AB} = \frac{DE}{BC} \]

We already found that:
\[ AB = \sqrt{3} \cdot AD \]

Thus:
\[ \frac{DE}{BC} = \frac{AD}{AB} = \frac{1}{\sqrt{3}} \]

Therefore:
\[ BC = \sqrt{3} \cdot DE \]

### Conclusion
We have shown that:
\[ \frac{AB}{AD} = \sqrt{3} \]
and
\[ BC = \sqrt{3} \cdot DE \]

Would you like more details on any specific step, or do you have any questions?

Here are some related questions you might be interested in:
1. How do you determine the similarity of triangles?
2. What are the properties of parallel lines in a triangle?
3. How do you find the ratio of areas of similar triangles?
4. What is the relationship between the sides of similar triangles?
5. How can you apply the Pythagorean Theorem in geometric proofs?

**Tip:** Always remember that the areas of similar triangles are proportional to the square of the corresponding sides.

Learn how to solve a geometry problem involving triangle areas, similarity, and proportions step-by-step. This problem explores the relationship between the areas of similar triangles and involves proving a geometric proportion. Suitable for high school students.

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