# Parallel lines & squares

A square $$P_1P_2P_3P_4$$ has points $$X$$ on side $$P_2P_3$$ and $$Y$$ on side $$P_3P_4$$ chosen such that angle $$XP_1Y$$ equals forty-five degrees. The lines $$P_1X$$ and $$P_1Y$$ intersect the circumcircle of the square at points $$R$$ and $$S$$, respectively.
How can one show that the lines $$XY$$ and $$RS$$ run parallel?

A geometric-algebraic solution using trigonometric addition properties is fairly straightforward and gives a solution. Yet here I´m looking for a more elegant elementary geometric solution. The intersections with the circumcircle, points $$R$$ and $$S$$ can be construed as corners of another square of equal size as the initial one using the Pythagorean theorem.
Here´s where I think a line or two or a smart angle hunt could solve it but got stuck. Any suggestion?

## locked by Pedro Tamaroff♦Feb 10 at 0:28

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