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This exercise involves vector analysis within a triangle and uses projections to determine relationships between segments. Let’s go through the problem step-by-step.

The problem specifies:
1. \( ABC \) is a triangle, and \( D \) is a point such that:
   \[
   \vec{CD} = \vec{AC} + 2 \vec{BC}
   \]
2. Points \( D_1 \) and \( D_2 \) are projections of \( D \) on lines \( (AB) \) and \( (AC) \), respectively, with the following instructions:

### Part 1
1. **a)** Show that \( \vec{AD_1} = -2 \vec{AB} \).
2. **b)** Show that \( \vec{AD_2} = 4 \vec{AC} \).

### Part 2
1. Construct a figure based on these points.
2. **a)** Show that \( \frac{\vec{AN}}{\vec{AB}} = \frac{2}{3} \).
3. **b)** Find the value of \( \frac{\vec{MN}}{\vec{MD}} \).

### Solution Steps

To proceed, let's first clarify a few steps and assumptions that might be useful:

1. **Vector Setup**:
   - Since \( \vec{CD} = \vec{AC} + 2 \vec{BC} \), you can use this to position \( D \) in terms of known vectors in the triangle.
   - Using the projection properties, apply geometric constraints on \( D_1 \) and \( D_2 \) to analyze the vectors along \( (AB) \) and \( (AC) \).

2. **Figure Construction**:
   - Construct the triangle \( ABC \), place \( D \) relative to points \( A \), \( B \), and \( C \), and mark points \( D_1 \), \( D_2 \), \( M \), and \( N \) accordingly.

Would you like a detailed step-by-step breakdown of each part, or is there a specific part you need more help with?

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Here are five related questions to deepen understanding:
1. What is the significance of vector projections in geometry?
2. How does the position of point \( D \) influence the projections onto \( (AB) \) and \( (AC) \)?
3. What role does the triangle’s orientation play in the interpretation of vector results?
4. How does the vector form of \( CD \) help determine the location of point \( D \) within the triangle?
5. What mathematical properties are used to confirm relationships like \( \vec{AD_1} = -2 \vec{AB} \)?

**Tip:** When working with vector projections, using coordinate geometry can simplify calculations by providing concrete representations of points.

ABC est un triangle et D est un point tel que: CD = AC + 2BC. Soit D1 le projeté de D sur (AB) parallèlement à (AC) et D2 le projeté de D sur (AC) parallèlement à (AB).

Explore vector projections and relationships in triangle ABC with point D. This problem includes vector addition and projection concepts, suitable for high school students studying geometry.

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