Inscribe Square in Circle using Compasses Alone

Problem

 * To inscribe a square in a given circle,
 * by means of compass alone,
 * supposing the center to be known.

Solution
Let $K$ denote the circle.


 * Inscribe-Square-in-Circle-with-Compasses.png

Locate point $B$ on the circumference at which one of the vertices of the square is to be located.

Construct $X$, $C$, $A$ from arcs of radius of $K$ center $B$, $X$ and $C$ respectively.

Construct arcs $CD$ and $XD$ with center $B$ and $A$ respectively of radius $BC$ and $AX$ respectively.

Construct $E$ and $F$ on the circumference of $K$ so as to make $AE$ and $AF$ the same length as $OD$.

$\Box AEBF$ is the square required.