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Transforming a right triangle by 90 degree rotations

Source: XI Olimpíada Matemática del Cono Sur (2000)

July 26, 2011
geometrygeometric transformationrotationanalytic geometryinvariantgeometry unsolved

Problem Statement

Consider the following transformation of the Cartesian plane: choose a lattice point and rotate the plane 90∘90^\circ counterclockwise about that lattice point. Is it possible, through a sequence of such transformations, to take the triangle with vertices (0,0)(0,0), (1,0)(1,0) and (0,1)(0,1) to the triangle with vertices (0,0)(0,0), (1,0)(1,0) and (1,1)(1,1)?
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