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?