the perpendicular bisector
The perpendicular bisector of a segment is the line that crosses the segment at its midpoint and stands at a right angle to it. It is the segment's line of mirror symmetry — fold the plane along it and the two endpoints land exactly on each other. Two conditions in one line: it bisects (cuts in half) and it is perpendicular.
Precisely: for a segment AB, its perpendicular bisector is the line through the midpoint M of AB that is perpendicular to AB. Its deep property is a locus characterization: a point P lies on the perpendicular bisector of AB if and only if it is equidistant from the two endpoints, |PA| = |PB|. One direction is quick — if P is on the perpendicular bisector, the two right triangles PMA and PMB are congruent by SAS, so |PA| = |PB| by CPCTC; the converse says any point equally far from A and B must sit on that line. This is why the perpendicular bisector is described as 'the set of all points equidistant from A and B'.
This locus idea is the seed of the circumscribed circle of a triangle: the center, equidistant from all three vertices, is found where the perpendicular bisectors of the three sides meet. Just remember the two requirements together — a line through the midpoint that is not perpendicular, or a perpendicular line that misses the midpoint, is not the perpendicular bisector and does not carry the equidistance property.
Segment AB has midpoint M. Any point P on the line through M perpendicular to AB satisfies |PA| = |PB|. So to find a point equally far from two towns, you stand somewhere on the perpendicular bisector of the segment joining them.
The perpendicular bisector is the locus of points equidistant from the endpoints.
Both conditions are required: through the midpoint AND perpendicular. The equidistance property is a theorem proven by congruent right triangles, not a definition.