STRAIGHT-LINE AND COMPASS CONSTRUCTIONS.

Problem 3

3. Given two lines that intersect at an angle, draw the line that cuts the angle of intersection precisely in half.

Solution

1. Let the two lines be A and B; let their point of intersection be c.
2. Draw a circle of arbitrary radius centred on c.
3. This circle intersects A and B twice each; pick a point of intersection with A (call it d) and a point of intersection with B (call it e) such that dae sweeps through the desired bisector.
4. Draw two new circles centred on d and e of radius de.
5. Let the points of intersection of these circles be f and g.
6. cfg are colinear and the line cfg is the required bisector of A and B. Note that selection of the other potential point e will construct the other bisector, which cuts this one at right angles.