Inscribe a square inside a circle

Let the centre of the circle be $O$ , and the vertex $A$ .

1. Draw a circle centred at $A$ passing through $O$ . Let the intersections with the circle be $P$ and $Q$ . (1 move)

2. Draw a circle centred at $P$ passing through $Q$ . Let it intersect the circle centred at $O$ again at $C$ . (2 moves)

3. Draw line $OP$ . Let it intersect the circle centred at $P$ at $E$ and $F$ . (3 moves)

4. Draw $CE$ and $CF$ . Let these lines intersect the circle centred at $O$ at $B$ and $D$ . (5 moves)

5. Draw $AB$ and $AD$ . (7 moves)

Here, $ABCD$ is the square we wished to produce, and it took 7 moves.