KöMaL 2019 B5067

The midpoint of side $AB$ of an acute angled triangle $ABC$ is $F$, and the line $e$ through the point $F$ halves the perimeter of triangle $ABC$. Line $e$ intersects the lines of sides $BC$ and $CA$ at points $D$ and $E$, respectively. Show that the perpendiculars drawn to $AB$ at $F$, to $BC$ at $D$, and to $CA$ at $E$ are concurrent.