Square with a Compass

Construct a square with only a compass and a straightedge.

Construct a 2 inch square with only a compass and a straightedge, with the
compass stuck at a distance of 2 inches.

Mark the four corners of a square using only a compass.
For each problem, try to use a minimum number of steps (lines, marks, etc.)
Source: rec.puzzles, Original, Henry Ernest Dudeny.
Solutions were received only from
Philippe Fondanaiche.
Feel free to keep sending solutions. Perhaps yours can improve upon these?

Draw a circle (c) with center O.
A diameter through O cuts (c) at the points A and B.
Draw the perpendicular bisector of AB which cuts (c) at the points C and D.
ABCD is a square.

Here is a solution giving a square of side 2 and a square of area 2.
Draw a circle (c) of center O and radius 2.
Draw a point Q on (c).
Determine on (c) the points J and K such as QJ = QK = 2.
Determine on (c) the point L such that JL=2.
The circles of centers J and L meet at O and O'.
OO' which is the perpendicular bisector of JL, cuts (c) at the point S.
Square of side 2:
Draw the circles of centers Q and S with radius 2.
They meet at a point T symmetrical of O about QS.
OSTQ is a square of side 2.
Square of area 2:
JK and QS meet at P.
Determine the point R on JK such as PR = 2.
OPQR is a square whose the diagonals are perpendicular and
equal to 2. The area of this square is equal to 2.

Draw a circle (c) with center A and radius 1.
Draw B on (c).
The circle of center B and radius 1, cuts (c) at C and D.
The circle of center C and radius 1, cuts (c) at B and E.
The circles of centers D and E with radius equal to DC = EB = sqrt(3)
meet themselves at F.
The circle of center D and radius AF = sqrt(2) cuts (c)
at G on the middle of the arch CB.
The circles of centers D and G and radius 1 cut at H.
ADHG is a square of side 1.
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