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4 tokens in the plane

Source: VI Caucasus Mathematical Olympiad

March 14, 2021
combinatorial geometrycombinatorics

Problem Statement

4 tokens are placed in the plane. If the tokens are now at the vertices of a convex quadrilateral PP, then the following move could be performed: choose one of the tokens and shift it in the direction perpendicular to the diagonal of PP 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\Pi, and after a number of moves tokens were at the vertices of one another rectangle Π′\Pi' such that Π′\Pi' is similar to Π\Pi but not equal to Π\Pi . Prove that Π\Pi is a square.
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