RO-NMO 2017 Problem 92

Let be a square $ABCD,$ a point $E$ on $AB,$ a point $N$ on $CD,$ points $F,M$ on $BC,$ name $P$ the intersection of $AN$ with $DE,$ and name $Q$ the intersection of $AM$ with $EF.$ If the triangles $AMN$ and $DEF$ are equilateral, prove that $PQ=FM.$