$M,N$ are midpoints of $AB$ and $CD$ for convex quadrilateral $ABCD$. Points $X$ and $Y$ are on $ AD$ and $BC$ and $XD=3AX,YC=3BY$. $\angle MXA=\angle MYB = 90$. Prove that $\angle XMN=\angle ABC$
Source: St Petersburg Olympiad 2010, Grade 10, P3
Tags: geometry
$M,N$ are midpoints of $AB$ and $CD$ for convex quadrilateral $ABCD$. Points $X$ and $Y$ are on $ AD$ and $BC$ and $XD=3AX,YC=3BY$. $\angle MXA=\angle MYB = 90$. Prove that $\angle XMN=\angle ABC$