Perpendicular bisector, step by step

Two equal arcs, one from each end, crossing above and below.

The question

Construct the bisector of AB.

compass+straightedge

Step 1→Open the compass to more than half of AB

Step 2→With the point on A, draw arcs above and below the segment

Step 3→Keeping the same opening, do the same from B

Step 4→Draw the line through the two crossings. It cuts AB in half, at a right angle

arcs=the evidence

Construct the bisector of AB. Two tools only — a compass that holds a distance and a straightedge that draws a line. Nothing here is measured.
1. Open the compass to more than half of AB. Any opening more than half of AB will do, and it has to be more than half or the two arcs never reach each other. This is the one condition in the whole construction.
2. With the point on A, draw arcs above and below the segment. Every point on this arc is the same distance from A. Hold that thought — it is the whole proof.
3. Keeping the same opening, do the same from B. The same opening from B means every point on the second arc is that same distance from B. So where the arcs cross, a point is equally far from A and from B.
4. Draw the line through the two crossings. It cuts AB in half, at a right angle. Two points equidistant from A and B determine the line of ALL such points, and that line is the perpendicular bisector. It cuts AB in half because the midpoint is on it, and it is perpendicular because the figure is symmetric about it.
Leave every arc showing. The arcs are the evidence the figure was constructed rather than eyeballed, and a marker reads them to check the method as well as the result.

How it works.

01
Construct the bisector of AB. Two tools only — a compass that holds a distance and a straightedge that draws a line. Nothing here is measured.
02
1. Open the compass to more than half of AB. Any opening more than half of AB will do, and it has to be more than half or the two arcs never reach each other. This is the one condition in the whole construction.
03
2. With the point on A, draw arcs above and below the segment. Every point on this arc is the same distance from A. Hold that thought — it is the whole proof.
04
3. Keeping the same opening, do the same from B. The same opening from B means every point on the second arc is that same distance from B. So where the arcs cross, a point is equally far from A and from B.
05
4. Draw the line through the two crossings. It cuts AB in half, at a right angle. Two points equidistant from A and B determine the line of ALL such points, and that line is the perpendicular bisector. It cuts AB in half because the midpoint is on it, and it is perpendicular because the figure is symmetric about it.
06
Leave every arc showing. The arcs are the evidence the figure was constructed rather than eyeballed, and a marker reads them to check the method as well as the result.