KöMaL 2019 B5016

In a convex quadrilateral $ABCD$, point $E_1$ lies on side $AD$, point $F_1$ lies on side $BC$, $E_2$ lies on diagonal $AC$, and $F_2$ lies on diagonal $BD$. Given that $AE_1:E_1D=BF_1:F_1C=AE_2:E_2C=BF_2:F_2D=AB:CD.$ and no pair of points coincide, prove that the lines $E_1F_1$ and $E_2F_2$ are perpendicular.