Introduction:

In geometry, 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 segment into two equal parts. To prove that every line segment has one and only one midpoint, we will need to provide a logical argument that supports this statement.

Proof:

1. Existence of a midpoint:
- Consider a line segment AB with endpoints A and B.
- To find the midpoint, draw a straight line from A to B.
- Let's call the point where the line intersects the segment AB as M.
- By construction, point M is the midpoint of segment AB.

2. Uniqueness of the midpoint:
- Assume there are two midpoints, M1 and M2, for the same line segment AB.
- Since both M1 and M2 are midpoints, they divide AB into two equal parts.
- Consider the segment AM1. Since M1 is the midpoint, AM1 is half the length of AB.
- Similarly, consider the segment AM2. Since M2 is the midpoint, AM2 is also half the length of AB.
- But segment AM1 and segment AM2 have the same length, as they are both half the length of AB.
- Therefore, AM1 is equal in length to AM2.
- Since A is the common endpoint for both segments, AM1 and AM2 overlap.
- This implies that M1 and M2 are the same point.
- Hence, there can only be one midpoint for a given line segment.

Conclusion:
- We have proved that every line segment has one and only one midpoint.
- This is based on the logical argument that a midpoint can be constructed by drawing a straight line through two endpoints, and that the uniqueness of the midpoint follows from the fact that two different midpoints would result in overlapping segments, which contradicts the definition of a midpoint.
- Therefore, the statement holds true for any line segment.