TITLE: construct a square without a ruler
QUESTION [11 upvotes]: 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.

REPLY [5 votes]: The key to solve this problem is how to construct $\sqrt{2}$.