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The image you uploaded contains a mathematical exercise focused on barycentric coordinates, geometric properties, and relationships. Let's break it down:

### Notation and Setup:
- \( G \) is defined as a barycentric combination of points \( A \), \( B \), and \( C \):
  \[
  G = \text{bary}\{ (A;2), (B;-2), (C;4) \}
  \]
  This indicates that point \( G \) is defined in terms of points \( A \), \( B \), and \( C \) with weights 2, -2, and 4, respectively.

### Questions Breakdown:
1. **Part 1:**
   - States that \( ABC \) is a parallelogram. Using this information might involve using the fact that for a parallelogram, opposite sides are equal and vectors behave in a specific manner. It could be relevant for verifying properties of \( G \).
   
2. **Part 2:**
   - Gives an equation involving \( G \):
     \[
     AB + 2AC = 0
     \]
     This likely represents a vector equation that simplifies the positions of the points.

3. **Part 3:**
   - Defines \( G \) with a different set of barycentric coordinates:
     \[
     G = \text{bary}\{ (A;1), (B;-2), (C;3) \}
     \]
     The new weights provide an alternative combination of points \( A \), \( B \), and \( C \) that forms point \( G \).

4. **Part 4:**
   - Introduces point \( D \), which is defined by the barycentric combination:
     \[
     D = \text{bary}\{ (A;0), (B;0), (C;1) \}
     \]
     and point \( E \):
     \[
     E = \text{bary}\{ (B;-2), (C;4) \}
     \]
     This seems to ask for relationships between these points and possibly a calculation of the midpoint or some other property.

5. **Part 5:**
   - Asks to identify which of the previous propositions are false.

### Next Steps:
- For each part, we would typically analyze the conditions given, work out the relationships (using the properties of barycentric coordinates), and verify any geometric or algebraic conditions (like verifying the vector equation).

Would you like to explore a specific part in more detail or need an explanation of how to work through barycentric coordinates?

### Follow-up Questions:
1. How do you determine the coordinates of a point given in barycentric form?
2. What is the significance of the vector equation \( AB + 2AC = 0 \) in the context of geometry?
3. How do you verify if \( ABC \) forms a parallelogram using vector properties?
4. What geometric interpretations can be made from the given barycentric coordinates for points \( D \) and \( E \)?
5. How would you approach checking which propositions are false?

### Tip:
When working with barycentric coordinates, remember that the sum of the weights doesn't necessarily have to equal 1, but it's a common condition for defining points relative to a triangle.

The image contains a math problem that involves barycentric coordinates and vector geometry. The user is asked to determine the barycentric combinations of points A, B, and C, and check whether certain propositions about a geometric figure (a parallelogram) are true or false.

This problem involves barycentric coordinates and vector geometry, exploring relationships between points A, B, and C, and verifying properties of a parallelogram. Learn how to solve vector equations and understand the geometric significance of barycentric combinations.

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