Problem

Source: Polish MO Second round 2024 P2

Tags: geometry



Let $ABCD$ be a convex quadrilateral with $\angle ABC=\angle ADC=120^{\circ}$. The point $E$ lies on the segment $AD$ and is such that $AE \cdot BC=AB \cdot DE$ and similarly the point $F$ lies on the segment $BC$ and satisfies $BF \cdot CD=AD \cdot CF$. Show that $BE$ and $DF$ are parallel.