I know Poncelet-Steiner tells us that given a circle and its center, straightedge alone is equivalent to straightedge and compass. My question is, what can we construct with purely straightedge? We certainly can't construct any square roots in a finite number of steps. Given a segment of unit length, is it possible to construct any rational number?
Thanks in advance. I wanted to know because I wanted to show that you can construct any square root with straightedge alone in an infinite number of steps.
EDIT: What would you need to construct every rational? Would some manner of constructing parallel lines suffice? Would a segment of length 2 in addition to the unit segment suffice?