1. Prove you can reflect points which lie on the sides of the square about the diagonals.
2. Construct two different rectangles whose vertices lie on the square and whose sides are parallel to the diagonals.
3. Construct points A, A', B, B' on one (extended) side of the square such that A/A' and B/B' are mirror image pairs with respect to another side of the square.
4. Construct the mirror image of the center of the square in one of the sides.
5. Divide the original square into 4 equal squares whose sides are parallel to the sides of the original square.
6. Divide one side of the square into 8 equal segments.
7. Construct a trapezoid in which one base is a square side and one base is 5/8 of the opposite square side.
8. Divide one side of the square into 5 equal segments.