Constructing a Square with a Given Diagonal

Construction Algorithm
1 Construct a line AC as a given diagonal of the square.
2 Construct M as a midpoint of the diagonal AC.
3 Construct the perpendicular line at M.
4 Define a rotation with center M and angle 90° using menu item "Mode/Transformation /Rotation".
5 Rotate the points A and C with the defined rotation to construct the points B and D.
6 Join up the four points A, B, C and D to get the required square.
7 Switch to drag mode. Pick a free element. Move it around to check your construction.
Construction Theorems
The two diagonals of the square are perpendicular to each other and equal in length.

Rotation preserves the distance between points.
Please enable Java for an interactive construction (with Cinderella).