Problem

Source: CGMO 2007 P5

Tags: geometry, circumcircle, geometric transformation, reflection, homothety, parallelogram, Hi



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 $DE \perp EF$.