Constructing a Perpendicular Bisector

Every point on a perpendicular bisector is the same distance from both endpoints of the segment. That's the whole idea behind the construction. If you swing an arc from endpoint A with some radius, every point on that arc is exactly that radius away from A. Swing an equal arc from endpoint B, and every point on it is that same radius away from B. Wherever the two arcs cross, you've found a point equidistant from A and B. Two such crossings, one above the segment and one below, both satisfy that condition, so the line joining them must be the set of all points equidistant from A and B — which is exactly the perpendicular bisector, passing through the midpoint at a right angle.

The compass opening just needs to be more than half the segment's length. Open it to less than half, and the two arcs are too short to reach each other — no crossing, no construction. Open it to anything past that halfway point, and the arcs always meet, because the radius only controls how far from the segment the crossings land, not whether they exist. Try a few different openings on paper and the bisector line comes out identical every time.

This same move appears again when you bisect all three sides of a triangle: the three perpendicular bisectors always meet at one point, called the circumcenter, which sits exactly as far from all three vertices. That point is the center of the circle that passes through all three corners of the triangle, tying this construction directly into circumscribed circles.