MathNet 2007-part-01 — 187 problems (EN–VI) Question 126

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 $DE \perp EF$.