Square with a Compass

  1. Construct a square with only a compass and a straightedge.
  2. Construct a 2 inch square with only a compass and a straightedge, with the compass stuck at a distance of 2 inches.
  3. 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?
  1. 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.

  2. 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.

  3. 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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