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The point cannot return to (1,sqrt2)

Source: Kürschák 1989, problem 3

July 20, 2014
analytic geometrycombinatorics unsolvedcombinatorics

Problem Statement

We play the following game in a Cartesian coordinate system in the plane. Given the input (x,y)(x,y), in one step, we may move to the point (x,y±2x)(x,y\pm 2x) or to the point (x±2y,y)(x\pm 2y,y). There is also an additional rule: it is not allowed to make two steps that lead back to the same point (i.e, to step backwards).
Prove that starting from the point (1;2)\left(1;\sqrt 2\right), we cannot return to it in finitely many steps.
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