You will need
Draw the sketch for the challenge. It is obvious that the diameter of the circle is the diagonal of the inscribed into this square circle. Remember a well-known property of the square: its diagonals are mutually perpendicular. Use this relationship of the diagonals when building a given square.
Draw the circle diameter. From the center by means of a square guide, a second diameter at an angle of 90 degrees to the first. Connect the points of intersection of the perpendicular diameters of the circle and get inscribed in this circle a square.
If the drawing tools you have only a compass, draw a circle. Mark on the circle an arbitrary point and swipe through her diameter with any object with a straight edge. Now you need with the help of a compass to divide a semi-circle between the ends of the diameter into two equal parts. From the points of intersection of the diameter with the circumference make two notches, keeping unchanged the solution of the compass. Through the point of intersection of these notches and the center of the circle do a second diameter. It is obvious that it will be perpendicular to the first.
If the drawing tools you have, scissors cut paper circle, outlined with a given circumference. Fold cut out a shape exactly in half. Retry the operation. To combine the ends of the bend line, then the curved portions coincide with no additional effort. Fix line addition. Now expand the circle. The fold lines are clearly visible. Fold the circle segments between points of intersection of the bend lines with the circle and cut these segments. Cut lines are the sides of the required square. Place the cut square in a given circle, aligning its center with the point of intersection of the fold lines of the circle. The vertices of the square will be lying on the circle that was required to perform.