Straightedge and compass - Hexagon

(1) Draw a circle radius r and center B.
(2) From a point A on the circumference of this circles, draw another circle radius r.
................Name the top point of intersection as C and the bottom as G.
(3) From B draw a line through A to intersect the left circle at E.
(4) With center E, radius r, draw a circle to intersect the left circle at D on top, F at bottom.
BCDEFG is the Hexagon. r=length of the side of the Hexagon.

I cannot prove that it cannot be constructed with less than 4 moves, but I can prove that 4 moves are enough.

1. Construct a circle $\omega_0$ centered at $O$ .
2. Pick a point on $\omega_0$ , call it $A$ . Construct a circle $\omega_1$ centered at $A$ with radius $AO$ .
3. Construct the line $AO$ . It intersects $\omega_0$ on $A$ and a new point, $D$ .
4. Construct a circle $\omega_2$ centered at $D$ with radius $DO$ .

Let the intersections of $\omega_0$ and $\omega_1$ be $B$ and $F$ . Let the intersections of $\omega_0$ and $\omega_2$ be $C$ and $E$ so $B$ and $C$ are on the same side of $AD$ . The hexagon is $ABCDEF$ .

Actually, we can do this in 4 steps with compass alone, no need for straightedge at all. I'll follow Ivan Koswara's diagram:

Step 1 and 2 same as Ivan's.

Step 3. Use B as center, BF as radius draw an arc, it'll intersect circle O at D

Step 4. Use A as center, BF as radius draw an arc, it'll intersect circle O at C and E

Now we are done!