Bisect a given line segment AB
Geometric Construction with the Compass Alone
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, ΔACE is similar to the ΔAFE (they are both isosceles and share ∠EAF. From here,