ELMO SL 2013 G6

Let $ABCDEF$ be a non-degenerate cyclic hexagon with no two opposite sides parallel, and define $X=AB\cap DE$ , $Y=BC\cap EF$ , and $Z=CD\cap FA$ . Prove that $$\frac{XY}{XZ}=\frac{BE}{AD}\frac{\sin |\angle{B}-\angle{E}|}{\sin |\angle{A}-\angle{D}|}.$$Proposed by Victor Wang