CGMO 2007 Problem 5

Point $D$ lies inside triangle $ABC$ such that $\angle DAC = \angle DCA = 30^{\circ}$ and $\angle DBA = 60^{\circ}$ . Point $E$ is the midpoint of segment $BC$ . Point $F$ lies on segment $AC$ with $AF = 2FC$ . Prove that $DE \perp EF$ .