If $T$ is the projection of the center of the circle on the line, then construct a square $ABCD$ with two vertices $A,B$ on the line s.t. $T$ is the midpoint of $AB$ and s.t. $C,D$ are on the same side of the line as the circle. Now draw lines $TA,TB$ . They intersect the circle in some points. Take $U\in TA,V\in TB$ to be two such points s.t. $UV$ is parallel to the line. Now let $P,Q$ be the projections of $U,V$ on the line. $PQVU$ is our square.