Diagonals of a rhombus
Assume two points A and B are given. A rhombus ABDC is to be constructed by following point C to be freely chosen under the restriction that the segments AB(=f) and AC(=g) are equal. That is, C is a circumpoint of the circle with center A and radius f, and D is an intersection of a circle c with center C and radius f, and a line h through C and parallel to f.
Are the diagonals of this rhombus perpendicular?
(This example was provided by Andreas Lindner.)