https://file.aiquickdraw.com/mathbot/file/1727347446150vu0l6rqa.jpg

https://file.aiquickdraw.com/mathbot/file/1727347446152b0puw16f.jpg

In the provided images, we have two elements:

1. **Textual information (problem statement):**
   - The problem states that point \( E \) is such that:
     \[
     \overrightarrow{AE} = \frac{1}{2} \overrightarrow{AB} + \frac{1}{2} \overrightarrow{CD} - \frac{1}{2} \overrightarrow{BD}
     \]
     It asks to show that the vectors \( \overrightarrow{AE} \), \( \overrightarrow{AB} \), and \( \overrightarrow{AC} \) are coplanar.

2. **Geometrical diagram:**
   - A tetrahedron with points \( A \), \( B \), \( C \), and \( D \) is illustrated. 

### Solution approach:

To show that the vectors \( \overrightarrow{AE} \), \( \overrightarrow{AB} \), and \( \overrightarrow{AC} \) are coplanar, we need to demonstrate that the scalar triple product of these vectors is zero:

\[
\overrightarrow{AE} \cdot (\overrightarrow{AB} \times \overrightarrow{AC}) = 0
\]

#### Step 1: Express \( \overrightarrow{AE} \)
Given in the problem:
\[
\overrightarrow{AE} = \frac{1}{2} \overrightarrow{AB} + \frac{1}{2} \overrightarrow{CD} - \frac{1}{2} \overrightarrow{BD}
\]
We simplify this by using the relation \( \overrightarrow{CD} = \overrightarrow{C} - \overrightarrow{D} \), etc., but we focus on understanding how this affects coplanarity.

#### Step 2: Calculate cross product \( \overrightarrow{AB} \times \overrightarrow{AC} \)
This gives the normal to the plane defined by points \( A \), \( B \), and \( C \).

#### Step 3: Compute the scalar triple product
Finally, check whether \( \overrightarrow{AE} \) is orthogonal to the normal vector, which ensures the vectors are coplanar.

Would you like me to proceed with detailed calculations?

Here are 5 related questions that could expand your understanding:

1. How is the scalar triple product used to determine coplanarity?
2. What is the geometric interpretation of vector addition in the equation for \( \overrightarrow{AE} \)?
3. How can we calculate the cross product of two vectors?
4. Why does the scalar triple product being zero confirm that vectors are coplanar?
5. What other methods can be used to prove vectors are coplanar?

**Tip:** Always remember that the scalar triple product can help in determining the volume of a parallelepiped, and if this volume is zero, it confirms that the vectors are coplanar.

Montrer que les vecteurs AE, AB, et AC sont coplanaires, sachant que AE = (1/2)AB + (1/2)CD - (1/2)BD.

Show that the vectors AE, AB, and AC are coplanar by using the scalar triple product. This problem covers vector addition and coplanarity conditions, ideal for high school students in Grades 10-12.

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