Perpendicular bisector
In a plane, the perpendicular bisector of a line segment is a line such that and are perpendicular and passes through the midpoint of .
In 3-D space, for each plane passing through that is not normal to there is a distinct perpendicular bisector. The set of lines which are perpendicular bisectors of form a plane which is the plane perpendicularly bisecting .
In a triangle, the perpendicular bisectors of all three sides intersect at the circumcenter.
Locus
The perpendicular bisector of is also the locus of points equidistant from and .
To prove this, we must prove that every point on the perpendicular bisector is equidistant from and , and also that every point equidistant from and .
The first part we prove as follows: Let be a point on the perpendicular bisector of , and let be the midpoint of . Then we observe that the (possibly degenerate) triangles and are congruent, by side-angle-side congruence. Hence the segments and are congruent, meaning that is equidistant from and .
To prove the second part, we let be any point equidistant from and , and we let be the midpoint of the segment . If and are the same point, then we are done. If and are not the same point, then we observe that the triangles and are congruent by side-side-side congruence, so the angles and are congruent. Since these angles are supplementary angles, each of them must be a right angle. Hence is the perpendicular bisector of , and we are done.