KöMaL 2019 B5040

In a square $ABCD$, let $F$ be an interior point of side $AB$, and let $E$ be an interior point of side $AD$. Draw a perpendicular to line $CE$ at point $E$, and a perpendicular to line $CF$ at point $F$. Denote the intersection of the two perpendiculars by $M$. Given that the area of triangle $CEF$ is half the area of pentagon $BCDEF$, prove that point $M$ lies on diagonal $AC$ of the square.