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A segment bisector or angle bisector. A bisector can be a line, line segment, or ray.
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.
The bisector and the line segment are perpendicular to each other.
By definition, a segment bisector always created two congruent segments.
A segment need not be a bisector. No theorem can be used to prove something that may not be true!