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The exercise provided is in French and involves vector relationships in a triangle geometry problem. Here’s a breakdown of the problem and how to approach it:

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Given:
- Triangle \( ABC \) with points \( E \) and \( F \) such that:
  \[
  \overrightarrow{CF} = \frac{1}{2} \overrightarrow{AB}
  \]
  and
  \[
  \overrightarrow{BE} = \frac{1}{2} \overrightarrow{AC}
  \]

Questions:
1. Draw the figure.
2. Show that:
   - \(\overrightarrow{AE} = \overrightarrow{AB} + 2 \overrightarrow{AC}\)
   - \(\overrightarrow{AF} = \overrightarrow{AB} + \overrightarrow{AC}\)
3. Deduce that points \( A \), \( E \), and \( F \) are aligned.

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### Solution Outline:

1. **Draw the Figure**: Construct a triangle \( ABC \). Then, place points \( E \) and \( F \) based on the given relationships:
   - \( F \) is along \( CF \) such that \( \overrightarrow{CF} = \frac{1}{2} \overrightarrow{AB} \).
   - \( E \) is along \( BE \) such that \( \overrightarrow{BE} = \frac{1}{2} \overrightarrow{AC} \).

2. **Proofs**:
   - **Show that** \( \overrightarrow{AE} = \overrightarrow{AB} + 2 \overrightarrow{AC} \):
     Using vector addition, and knowing \( \overrightarrow{BE} = \frac{1}{2} \overrightarrow{AC} \), you can express \( \overrightarrow{AE} \) as the sum of vectors \( \overrightarrow{AB} \) and \( 2 \overrightarrow{AC} \).
   
   - **Show that** \( \overrightarrow{AF} = \overrightarrow{AB} + \overrightarrow{AC} \):
     Similarly, express \( \overrightarrow{AF} \) as a sum of \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \).

3. **Alignment of Points**:
   - To show \( A \), \( E \), and \( F \) are collinear, verify that \( \overrightarrow{AE} \) and \( \overrightarrow{AF} \) are scalar multiples of each other, implying they lie on the same line.

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Would you like me to expand on any specific step in this solution, or provide detailed calculations?

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### Related Questions:

1. How would the solution change if the vectors \( \overrightarrow{CF} \) and \( \overrightarrow{BE} \) had different coefficients?
2. What is the significance of collinearity in vector geometry?
3. How do you determine if three points are collinear using vectors?
4. Can we solve this problem using coordinates instead of vectors?
5. How would this problem change if \( E \) and \( F \) were defined differently?

**Tip:** In vector proofs, breaking down vectors into smaller components or using properties like scalar multiplication often simplifies the work.

In triangle ABC, two points E and F are considered such that CF = 1/2 AB and BE = 1/2 AC. Prove that AE = AB + 2AC and AF = AB + AC, then deduce that points A, E, and F are aligned.

Learn how to prove the alignment of points in triangle ABC using vector relationships. This problem involves calculating vectors AE and AF, and demonstrating that points A, E, and F are collinear. Suitable for high school students studying vector geometry.

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