IMO 2002 — Complete Shortlist Question 8

Let $ABC$ be a triangle for which there exists an interior point $F$ such that $\angle AFB = \angle BFC = \angle CFA$. Let the lines $BF$ and $CF$ meet the sides $AC$ and $AB$ at $D$ and $E$ respectively. Prove that

$$ AB\;+\;AC\;\geq\;4DE. $$