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This problem is about proving the relationship between the segments in a triangle when a line intersects the sides and is parallel to the third side. Here's a breakdown of the proof:

### Problem:
Given a triangle \( \triangle ABC \), a line intersects sides \( AB \) and \( AC \) at points \( D \) and \( E \) respectively, and this line is parallel to side \( BC \). We need to prove that:
\[
\frac{AD}{AB} = \frac{AE}{AC}
\]
This is related to the Basic Proportionality Theorem (also called Thales' Theorem).

### Solution:

1. **Given**: 
   The line \( DE \parallel BC \), and it intersects \( AB \) at \( D \) and \( AC \) at \( E \).

2. **By the Basic Proportionality Theorem** (Theorem 6.1 as mentioned):
   Since \( DE \parallel BC \), it follows that:
   \[
   \frac{AD}{DB} = \frac{AE}{EC}
   \]

3. **Rearrange this equation** to express it differently:
   \[
   \frac{DB}{AD} = \frac{EC}{AE}
   \]

4. **Adding 1 on both sides**:
   \[
   \frac{DB}{AD} + 1 = \frac{EC}{AE} + 1
   \]
   This simplifies to:
   \[
   \frac{AB}{AD} = \frac{AC}{AE}
   \]
   (since \( AB = AD + DB \) and \( AC = AE + EC \)).

5. **Taking the reciprocal** on both sides gives the desired result:
   \[
   \frac{AD}{AB} = \frac{AE}{AC}
   \]
   This completes the proof.

This is a direct application of the Basic Proportionality Theorem, where the line parallel to one side of a triangle divides the other two sides in the same ratio.

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Would you like a more detailed explanation of any step?

Here are 5 related questions to deepen your understanding:
1. How is the Basic Proportionality Theorem applied in different triangle configurations?
2. Can this theorem be extended to other polygons or non-Euclidean geometry?
3. What happens if the line is not parallel to side \( BC \)?
4. How can this theorem help in solving real-life geometric problems?
5. How would the proof change if instead of proving \(\frac{AD}{AB} = \frac{AE}{AC}\), we had to find the length of one segment given others?

### Tip: 
The Basic Proportionality Theorem is fundamental for many geometric constructions and proofs, especially in coordinate geometry and similarity theorems.

If a line intersects sides AB and AC of a triangle ABC at D and E respectively and is parallel to BC, prove that AD/AB = AE/AC.

Learn how to prove that AD/AB = AE/AC in a triangle where a line parallel to one side intersects the other two sides. This problem covers the Basic Proportionality Theorem and its applications. Suitable for students in Grades 9-10.

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