4 tokens are placed in the plane. If the tokens are now at the vertices of a convex quadrilateral P, then the following move could be performed: choose one of the tokens and shift it in the direction perpendicular to the diagonal of P not containing this token; while shifting tokens it is prohibited to get three collinear tokens. Suppose that initially tokens were at the vertices of a rectangle Π, and after a number of moves tokens were at the vertices of one another rectangle Π′ such that Π′ is similar to Π but not equal to Π. Prove that Π is a square.
combinatorial geometrycombinatorics