Proof that a Line Segment has only one Midpoint:

A line segment is a part of a line that is bounded by two distinct endpoints. The midpoint of a line segment is the point that divides the line segment into two equal parts.

To prove that a line segment has only one midpoint, we can use a proof by contradiction. Let's assume that a line segment has two distinct midpoints, M1 and M2.

Proof by Contradiction:

1. Assume that there are two distinct midpoints, M1 and M2:
- Let's assume that the line segment AB has two midpoints, M1 and M2.
- Therefore, AM1 = M1B and AM2 = M2B.

2. Show that M1 and M2 are coincident:
- Since M1 and M2 are midpoints, they divide the line segment AB into two equal parts.
- Therefore, AM1 = AM2 and M1B = M2B.

3. Use the Transitive Property of Equality:
- According to the Transitive Property of Equality, if two things are equal to the same thing, then they are equal to each other.
- Since AM1 = AM2 and AM1 = M1B, we can conclude that AM2 = M1B.

4. Apply the Segment Addition Postulate:
- The Segment Addition Postulate states that if three points A, M, and B are collinear, then AM + MB = AB.
- Applying this postulate, we have AM2 + M1B = AM1 + M1B = AB.

5. Simplify the equation:
- Since AM2 = M1B, the equation becomes AM2 + AM2 = AB.
- Simplifying further, we get 2AM2 = AB.

6. Show that M2 is the midpoint of AB:
- From the equation 2AM2 = AB, we can divide both sides by 2 to get AM2 = AB/2.
- This means that AM2 is half the length of AB.
- Therefore, M2 divides AB into two equal parts, making it the midpoint.

7. Contradiction:
- We assumed that M1 and M2 are distinct midpoints, but we have shown that M2 is the midpoint.
- This contradicts our initial assumption, proving that a line segment has only one midpoint.

Conclusion:
By using a proof by contradiction, we have shown that a line segment cannot have two distinct midpoints. Therefore, a line segment has only one midpoint.