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No, the definition of a bisector is the point at which a segment is divided into two equal halves. Of course, a segment may be divided further. However, there can be only one bisector of any one segment.
Line segment
That means that it is not a line segment.
There is only one point on the line segment, which is equidistant from the endpoints.
Line segment because a line is endless and a segment is not a segment has to extremites and so is the book...