ÖMO 2019 Problem 5

Let $ABC$ be an acute-angled triangle. Let $D$ and $E$ be the feet of the altitudes on the sides $BC$ or $AC$ . Points $F$ and $G$ are located on the lines $AD$ and $BE$ in such a way that $\frac{AF}{FD}=\frac{BG}{GE}$ . The line passing through $C$ and $F$ intersects $BE$ at point $H$ , and the line passing through $C$ and $G$ intersects $AD$ at point $I$ . Prove that points $F, G, H$ and $I$ lie on a circle.

(Walther Janous)