construct a square without a ruler [duplicate]

How can I construct a square using only a pencil and a compass, i.e. no ruler.

Given: a sheet of paper with $2$ points marked on it, a pencil and a compass.

Aim: plot $2$ vertices such that the $4$ of them form a square using only a compass.

P.S.: no cheap tricks involved.

Solution 1:

On way to do this is to start by constructing the middle point of your segment like this:

Then you will easily have the middle of your square like this:

And you know can have a circonscrit circle for your square, and constructing the square is finally possible!

Edit.

Since you asked for it, I have made a few more drawings to illustrate how to construct the point $O$ from where we left it.

Where the last circle has for radius $CF$ and for center $A$.

Edit 2.

Since more details were requested, here is how to finish the proof once $O$ has been constructed.

Solution 2:

The key to solve this problem is how to construct $\sqrt{2}$.