A quadrilateral is rotated clockwise, and the sides are extended its length in the direction of the movement. It turns out the endpoints of the segments constructed form a square. Prove the initial quadrilateral must also be a square. Generalization: Prove that if the same process is applied to any $n$-gon and the result is a regular $n$-gon, then the intial $n$-gon must also be regular.