The ancient Greeks felt that the line and the circle are the basic figures and the straightedge and compass are their physical analogues. Hence they were interested in what can be done with a straightedge and compass, i.e. what figures are Euclidean constructions. For example, given a line l and a point P on l, it is possible to construct the perpendicular to l though P using only a straightedge and compass as follows: with the compass draw a circle C0 with center P. Let A and B be the intersection points of l with C0. Next draw the circles with centers A and B which pass through B and A respectively. These two circles intersect in two points E and F say. The line segment EF is perpendicular to l.

If one is given segments AB and CD of lengths x and y respectively...