Question 18
Interior points $D$, $E$, $F$ are chosen on the sides $BC$, $CA$, $AB$ (not at the vertices). Let $k$ be the length of the longest side of $DEF$. Let $a$, $b$, $c$ be the lengths of the longest sides of $AFE$, $BDF$, $CDE$ respectively. Show that $k \geq \dfrac{\sqrt{3}}{2} \min(a, b, c)$. When do we have equality?