CAPS 2017 Problem 2

Let ${\omega}$ be the circumcircle of an acute-angled triangle ${ABC}$ . Point ${D}$ lies on the arc ${BC}$ of ${\omega}$ not containing point ${A}$ . Point ${E}$ lies in the interior of the triangle ${ABC}$ , does not lie on the line ${AD}$ , and satisfies ${\angle DBE =\angle ACB}$ and ${\angle DCE = \angle ABC}$ . Let ${F}$ be a point on the line ${AD}$ such that lines ${EF}$ and ${BC}$ are parallel, and let ${G}$ be a point on ${\omega}$ different from ${A}$ such that ${AF = FG}$ . Prove that points ${D,E, F,G}$ lie on one circle.

(Slovakia)