GeoProof — Perpendicular Bisector explorer

P is on the perpendicular bisector ⟺ PA = PB

The Perpendicular Bisector

The perpendicular bisector of a segment AB is the line that passes through the segment's midpoint at a right angle. Its defining property is about distance:

a point is equidistant from A and B ⟺ it lies on the perpendicular bisector

So the perpendicular bisector is exactly the set (the "locus") of all points the same distance from A as from B.

Both directions

If P is on the line, then PA = PB (drag P along it and watch the two distances stay equal). And conversely, any point with PA = PB has to be on that line — there is nowhere else to be equidistant.

Why it's true

Joining P to the midpoint M of AB makes two right triangles that share the side PM and have equal halves MA = MB. They are congruent, so PA = PB. Run the argument backwards for the converse.

Where it's used

Three perpendicular bisectors of a triangle's sides meet at the circumcenter — the point equidistant from all three vertices. The idea also drives "nearest-point" partitions (Voronoi diagrams) and many constructions.