Geometric Construction with the Compass Alone

Build a square with a given side AB.

Solution

Using Problem #4 first construct CA perpendicular to AB. Draw the circle A(B) of radius AB and center A and another centered at C and intersecting the former in points, say, M and N. We are given then circle A(B) and an arc MN that can be bisected following the construction in Problem #7. Both thus found points will lie on AC at the distance from A equal to AB. Denote one of them D: AD = AB and BAD = 90°.

Swing two arcs of radius AB - one centered at D, another at B. The arcs will intersect at A and another point, say, E. The quadrilateral ADEB is a square.