Question 35
$ABC$ is an acute angled triangle. The midpoints of $BC$, $CA$ and $AB$ are $D$, $E$, $F$ respectively. Perpendiculars are drawn from $D$ to $AB$ and $CA$, from $E$ to $BC$ and $AB$, and from $F$ to $CA$ and $BC$. The perpendiculars form a hexagon. Show that its area is half the area of the triangle.