No, because if there is more than one midpoint then it will be like multiple line segments connected together. You can't have two midpoints or more, but you can have one to divide the line segment in half.
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In normal metric spaces a line segment cannot have two mid points.
The centre of a line segment
It is the mid-point of a line segment that has end points
Mid line segment.
No. Since a line is infinite, it has no mid-point. A bisector must go through a midpoint so nothing can bisect a line (not even a segment).