CAPS 2017 Problem 4

Let ${ABC}$ be a triangle. Line l is parallel to ${BC}$ and it respectively intersects side ${AB}$ at point ${D}$ , side ${AC}$ at point ${E}$ , and the circumcircle of the triangle ${ABC}$ at points ${F}$ and ${G}$ , where points ${F,D,E,G}$ lie in this order on l. The circumcircles of triangles ${FEB}$ and ${DGC}$ intersect at points ${P}$ and ${Q}$ . Prove that points ${A, P,Q}$ are collinear.

(Slovakia)