Draw a Square Without a Compass, Only a Straightedge

Question

I remember seeing the following question in an old STEP question:

using only a straight-edge and a set of (unmarked) coordinate axes, construct a square.

I'm sure I knew how to do it when I was preparing for STEP, but I don't remember now. I also can't find the STEP question, otherwise there would likely be a solution for it on The Student Room. I just remember it being pretty cool!

---

Update. So it seems that this problem is not solvable (and hence I misremembered the question). Having given it some further thought, I think I have remembered the question correctly now; I have reposted it at Draw a Square Without a Compass, Only a Straightedge -- Part Deux. (I thought it seemed too large a change to be ok as an edit, as all the comments and answers would no longer be relevant.)

Answer 1

How about joining $\left(1,0\right)$ to $\left(0,1\right)$ to $\left(-1,0\right)$ to $\left(0,-1\right)$ back to $\left(1,0\right)$ by straight lines?

Answer 2

Not that I've proved it, but I suspect this is impossible.

That's assuming that "using a straightedge" means exactly what it means in traditional Euclidean constructions.

Just to inject a minimal amount of content, I also have a conjecture how this is consistent with the fact that the OP says he saw a solution once. I conjecture that that solution cheated. For example, it's easy to construct a square if we're given an actual physical ruler and we're allowed to use the fact that the two edges are parallel.

And a vague notion of where a proof of impossibility might come from: The constructions we can make with a straightedge, starting with those two axes, are invariant under the transformation $(x,y)\mapsto(2x,y)$.