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OMMC POTM, 2022 10

Define a convex quadrilateral $\mathcal{P}$ on the plane. In a turn, it is allowed to take some vertex of $\mathcal{P}$, move it perpendicular to the current diagonal of $\mathcal{P}$ not containing it, so long as it never crosses that diagonal. Initially $\mathcal{P}$ is a parallelogram and after several turns, it is similar but not congruent to its original shape. Show that $\mathcal P$ is a rhombus. [i]Proposed by Evan Chang (squareman), USA[/i]

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