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The problem in the image asks to prove that the midpoint \( M \) of the line segment joining two points \( P_1(x_1, y_1, z_1) \) and \( P_2(x_2, y_2, z_2) \) is given by the coordinates:

\[
M \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right)
\]

### Proof:

To find the midpoint of a line segment between two points in three-dimensional space, we use the **midpoint formula**. This formula states that the midpoint \( M(x_m, y_m, z_m) \) between two points \( P_1(x_1, y_1, z_1) \) and \( P_2(x_2, y_2, z_2) \) is the average of the corresponding coordinates of the two points.

#### 1. Formula for the midpoint:
The coordinates of \( M \) can be calculated as follows:

\[
x_m = \frac{x_1 + x_2}{2}, \quad y_m = \frac{y_1 + y_2}{2}, \quad z_m = \frac{z_1 + z_2}{2}
\]

#### 2. Explanation:
- For the **x-coordinate**, we average the x-coordinates of \( P_1 \) and \( P_2 \):
  \[
  x_m = \frac{x_1 + x_2}{2}
  \]
  
- For the **y-coordinate**, we average the y-coordinates of \( P_1 \) and \( P_2 \):
  \[
  y_m = \frac{y_1 + y_2}{2}
  \]
  
- For the **z-coordinate**, we average the z-coordinates of \( P_1 \) and \( P_2 \):
  \[
  z_m = \frac{z_1 + z_2}{2}
  \]

Thus, the midpoint \( M \) is located at:

\[
M \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right)
\]

This matches the given expression in the problem, and hence the proof is complete.

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Would you like further details or clarifications on any specific part of this proof?

### Follow-up questions:
1. What is the formula for the midpoint of a line segment in 2D?
2. How does the midpoint formula extend to higher dimensions?
3. Can you derive the midpoint formula using vectors?
4. How can the midpoint formula be used in geometric applications, like finding centers of figures?
5. Can the midpoint formula be applied to non-Euclidean geometries?

#### Tip:
When finding midpoints in geometry, always remember to average the coordinates of the endpoints of the line segment!

Prove that the midpoint M of the line segment joining points P1(x1, y1, z1) and P2(x2, y2, z2) is the point ( (x1 + x2)/2, (y1 + y2)/2, (z1 + z2)/2 ).

This page explains how to prove the midpoint formula for a line segment in 3D space. The proof covers key concepts in coordinate geometry and uses the midpoint theorem to show that the midpoint M of the line segment joining points P1(x1, y1, z1) and P2(x2, y2, z2) is the point ( (x1 + x2)/2, (y1 + y2)/2, (z1 + z2)/2 ).

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