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The two points are exactly the same so there is no line segment.
All bisectors intersect the line segment at the midpoint. There can be multiple bisectors, intersecting at the midpoint at different angles, but they all intersect the line segment at its midpoint. The midpoint separates the line segment into two equal halves.
Perpendicular Bisector
Take a compass, extend it about 3/4 of the length of the segment. Then from one end of the segment, draw a 180 degree arc. From the other end draw another arc. Connect the points where the arcs intersect. Where the line intersects with the segment is the midpoint of the segment. That is how you bisect a segment to find the midpoint - geometrically.
it depends on how long or how many joining segments it has. normally one line segment contains only one midpoint. Unless it has a joining segment there is only one midpoint.