I am hoping that the brilliant MathOverflow geometers can help me out.
Question 1. Suppose that I have a fixed finite-length straightedge and fixed finite-size compass. Can I still construct all constructible points in the plane?
I know the answers in several variations.
If I have compasses of arbitrary size, then I don't need a straightedge at all. This is the Mohr–Moscheroni compass-only theorem.
If I have an infinite straightedge (or arbitrarily large), then I need only a single compass of any fixed size. This is the rusty compass theorem.
Indeed, the Poncelet–Steiner theorem shows that I need only an infinite straightedge and a single circle of known center and radius.
But what I don't know is the remaining case, where both the straightedge and compass are limited in size. The difficult case seems to be where you have two points very far apart and you want to construct the line joining them.
Will Sawin's comment answers the question I had asked above. But it doesn't seem to answer the relative version of the question, which is what I had had in mind:
Question 2. Suppose that I have a fixed finite-length straightedge and fixed finite-size compass. Can I still construct all constructible points in the plane, relative to a fixed finite set of points?
In other words, is the tool set of a finite straightedge and finite compass fully equivalent to the arbitrary size toolset we usually think about.