construct a square without a ruler 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.
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.
The only difficulty is to create Perpendicular lines.
One possible way could be, if you are precise enough:

lets suppose the vertices are (P1,P2,P3,P4), where P1 and P2 are given two points.

Join the two points(P1,P2) and extend on both sides.

To draw a perpendicular at point(P1) on the line: Mark two points on the line equidistant from P1 on both sides, then place your compass at the marked points, there after create arcs using your compass such that they intersect.

joining the intersection of arcs and P1 would give you a perpendicular line on the given line.

Since you have created a perpendicular line, then it's not difficult to create a Square from it.