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The problem you're working with seems to involve affine geometry and transformations, specifically central symmetries in an affine space. Here's a breakdown of the problem from what I can read:

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Given:
- \( A \) and \( B \) are two distinct points in an affine space.
- \( s_A \) and \( s_B \) represent the central symmetries centered at points \( A \) and \( B \), respectively.
- \( \vec{u} \) is the vector \( \overrightarrow{AB} \).

The question asks which of the four statements is correct:
- A) \( S_A \circ t_{\vec{u}} = t_{\vec{u}} \circ S_A \)
- B) \( S_A \circ S_B = S_B \circ S_A \)
- C) \( S_A \circ t_{\vec{u}} \circ S_A = t_{-\vec{u}} \)
- D) \( S_A \circ S_B = t_{2\vec{u}} \)

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To solve this, let's first break down the components:

1. **Central symmetry** \( S_A \): This transformation reflects a point \( P \) with respect to the center \( A \). The point \( P \) is mapped to the point \( P' \) such that \( \overrightarrow{AP'} = -\overrightarrow{AP} \).
   
2. **Translation** \( t_{\vec{u}} \): This transformation translates a point \( P \) by the vector \( \vec{u} = \overrightarrow{AB} \), meaning \( P \) is mapped to \( P + \vec{u} \).

Now, let's examine each of the proposed equations:

- **Option A**: \( S_A \circ t_{\vec{u}} = t_{\vec{u}} \circ S_A \)
    - This is asking whether applying a translation by \( \vec{u} \) and then applying the symmetry around \( A \) is the same as applying the symmetry first and then the translation. This would only be true if the operations commute, which they generally do not because symmetries and translations affect points differently.

- **Option B**: \( S_A \circ S_B = S_B \circ S_A \)
    - This is asking whether symmetries around different points commute. Since \( S_A \) and \( S_B \) are reflections with respect to two distinct points, they generally do not commute. The order of applying these transformations matters.

- **Option C**: \( S_A \circ t_{\vec{u}} \circ S_A = t_{-\vec{u}} \)
    - This is plausible. If you first apply the symmetry around \( A \), then a translation by \( \vec{u} \), and then another symmetry around \( A \), the effect should be equivalent to a translation by the opposite vector \( -\vec{u} \), because the two symmetries "reverse" the translation.

- **Option D**: \( S_A \circ S_B = t_{2\vec{u}} \)
    - This seems likely. If you apply a reflection around \( A \), then a reflection around \( B \), the result is a translation by twice the vector \( \vec{u} \), because the point is first reflected over \( A \), then moved twice the distance between \( A \) and \( B \).

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Thus, based on the reasoning above:

- **Option C** and **Option D** are the strongest candidates. 
- **Option C** is likely correct since the combination of symmetry and translation tends to invert the direction of the translation.
- **Option D** might also hold because consecutive symmetries result in a translation.

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Would you like me to go deeper into the explanation of any of these concepts? Here are 5 related questions that might help solidify understanding:

1. What are the properties of central symmetries in affine spaces?
2. How do translations and symmetries interact geometrically?
3. Can you prove that symmetries do not commute in general?
4. How does the composition of two central symmetries lead to a translation?
5. How does symmetry affect the vector of translation in affine transformations?

**Tip:** When combining affine transformations (like symmetry and translation), it is helpful to visualize the action on a specific point to understand their combined effect.

Soient A et B deux points distincts d’un espace affine, sA et sB les symétries centrales de centres A et B, et u⃗ = AB. On a :
 A) SA ∘ tu = tu ∘ SA
 B) SA ∘ SB = SB ∘ SA
 C) SA ∘ tu ∘ SA = t-u
 D) SA ∘ SB = t2u

Explore this advanced problem in affine geometry involving central symmetries and translations. Learn how to solve for symmetries at distinct points A and B, with step-by-step analysis of transformations. Suitable for university-level students studying geometry or linear algebra.

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