Geometric Construction with the Compass Alone

Bisect a given line AB.

Solution

Construct a point C such that AC = 2·AB (Problem #1). Swing an arc of radius CA with center C. This will intersect the circle of radius AB centered at A at points D and E. Draw two arcs of radii DA and EA (which are both equal to AB) and centers at, respectively, D and E. Besides A, the two arcs will intersect at another point F which is the middle of the segment AB.

Indeed, ACE is similar to the Δ&bsp;AFE (they are both isosceles and share EAF. From here, AC/AE = AE/AF. But AC/AE = 2·AB/AB = 2. Therefore, AB = AE = 2·AF.