Constructing Angle Bisectors and Perpendicular Bisectors

The angle bisector construction divides an angle into two equal parts, and the perpendicular bisector construction finds the midpoint of a segment and the line perpendicular to it, both with a compass and a straightedge. In brief: for a perpendicular bisector draw two equal circles centered at the ends of the segment and join their intersection points; for an angle bisector mark equal distances on both sides of the angle and draw equal arcs from those marks. Each construction can be justified by congruent triangles.

The two constructions are the basis of many others: finding the center of a circle through three points, the incenter and circumcenter of a triangle, and the construction of angles such as 30°, 45° and 22.5°. This guide gives the steps, the proofs that they work, and applications including the incircle and circumcircle of a triangle.

How do you construct a perpendicular bisector?

1. Draw the segment AB.
2. Set the compass to a radius greater than half of AB. Draw an arc above and below the segment with center A.
3. With the same radius, draw an arc with center B that crosses the first arcs at two points, P and Q.
4. Draw the line PQ. It meets AB at its midpoint M and is perpendicular to AB.

Every point on the perpendicular bisector is equally far from A and from B. This property is useful: the set of points equidistant from two points is exactly the perpendicular bisector.

How do you construct an angle bisector?

1. Let the angle have vertex V and sides s₁ and s₂.
2. Draw an arc with center V that cuts the sides at P and Q.
3. With the same radius (greater than half of PQ), draw arcs centered at P and Q that meet at a point R inside the angle.
4. Draw the ray VR. It bisects the angle.

Every point on the angle bisector is equally far from the two sides, which makes it the locus of points equidistant from two intersecting lines.

Why do the constructions work?

Perpendicular bisector. Since the circles have equal radii r, AP = BP = AQ = BQ = r. Thus triangles APQ and BPQ are congruent (SSS), and the quadrilateral APBQ is a rhombus. The diagonals of a rhombus are perpendicular and bisect each other, so PQ ⟂ AB and M is the midpoint.

Angle bisector. VP = VQ (radius of the first arc) and PR = QR (equal radii of the second arcs), and VR is common. So triangles VPR and VQR are congruent by SSS, and the corresponding angles ∠PVR and ∠QVR are equal.

What can you build with them?

Center of a circle through three points

Construct the perpendicular bisectors of two sides of the triangle formed by the points. They meet at the circumcenter, which is equally far from the three vertices. This point is the center of the circle through all three points. For a given arc, this method finds the center, which is useful for restoring a broken circular object.

The incircle

The bisectors of the three angles of a triangle meet at the incenter, which is equally far from the three sides. Drop a perpendicular from the incenter to a side to get the radius of the incircle.

Angles by bisection

Bisecting 60° gives 30°. Bisecting 90° gives 45°. Adding 60° and 30° gives 90°, and adding 60° and 45° gives 105°.

Constructible angles from bisection

A general angle cannot be trisected with compass and straightedge, as proved in the nineteenth century. For example, 20° is not constructible, because it would trisect 60°.

Where do these constructions come from?

Both constructions appear in Euclid’s Elements in Book I, Propositions 9 and 10 for the angle bisector and the midpoint, and Proposition 11 and 12 for perpendiculars. They were the first constructions to be proved with the congruence of triangles. The idea of the circumcircle and incircle was developed by Greek geometers and appears in Book IV of the Elements.

Where are they used today?

Craftspeople use them for layout, such as finding the center of a board and bisecting angles for mitred joints. Surveyors use perpendicular bisectors to find positions equally far from two landmarks. In computer graphics and robotics, the perpendicular bisector defines regions of closeness to two points, which is the idea behind Voronoi diagrams.

Which mistakes are common?

Using a radius that is too small

For the perpendicular bisector the radius must be larger than half of AB, or the arcs will not meet.

Different radii for the two arcs

The arcs from the two endpoints must have the same radius. Otherwise the line is not the bisector.

Confusing the median with the bisector

The perpendicular bisector of a side of a triangle does not generally pass through the opposite vertex. Only in an isosceles triangle, for the base, do the lines coincide.

Stopping the angle bisector at the arc

Draw the ray from the vertex through the point R, and check the result by measuring the two halves as a final test.