Question 57
Let $M$ be the midpoint of side $BC$ of acute-angled triangle $ABC$, and let $E$ and $F$ be the feet of the perpendiculars from $M$ to sides $AC$ and $AB$, respectively. Points $X$ and $Y$ are such that $\triangle CEY \sim \triangle XEC$ and $\triangle XBF \sim \triangle BYF$ (the corresponding vertices of the triangles are in the same order as written), and points $E$ and $F$ are not located on line $XY$. Prove that $AM \perp XY$.