Bisect an angle, step by step

Cut an angle exactly in half with three arcs and a straightedge.

The question

Bisect angle ABC.

compass+straightedge

Step 1→With the point on B, draw an arc crossing both arms. Call the crossings P and Q

Step 2→With the point on P, draw an arc inside the angle

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

Step 4→Draw the line from B through where the two arcs cross. It halves the angle

arcs=the evidence

Bisect angle ABC. Two tools only — a compass that holds a distance and a straightedge that draws a line. Nothing here is measured.
1. With the point on B, draw an arc crossing both arms. Call the crossings P and Q. The arc turns the two arms into two points, P and Q, the same distance from B. The angle is now a pair of points rather than an opening.
2. With the point on P, draw an arc inside the angle. The arc from P holds every point at one distance from P.
3. Keeping the same opening, do the same from Q. The same opening from Q does the same for Q, so their crossing is equally far from P and from Q.
4. Draw the line from B through where the two arcs cross. It halves the angle. B is equally far from P and Q as well, so the line from B through the crossing is the axis of symmetry of the whole figure — which is exactly what halving the angle means.
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
Bisect angle ABC. Two tools only — a compass that holds a distance and a straightedge that draws a line. Nothing here is measured.
02
1. With the point on B, draw an arc crossing both arms. Call the crossings P and Q. The arc turns the two arms into two points, P and Q, the same distance from B. The angle is now a pair of points rather than an opening.
03
2. With the point on P, draw an arc inside the angle. The arc from P holds every point at one distance from P.
04
3. Keeping the same opening, do the same from Q. The same opening from Q does the same for Q, so their crossing is equally far from P and from Q.
05
4. Draw the line from B through where the two arcs cross. It halves the angle. B is equally far from P and Q as well, so the line from B through the crossing is the axis of symmetry of the whole figure — which is exactly what halving the angle means.
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.