Easy problem. Here's my proof. Proof: Firstly notice that $DG=AH \implies EG=HB$. Also $EA=BC$ and $$\angle GEA=\angle HBC \implies \Delta GEA \cong \Delta HBC\implies GA=HC$$. Also this implies $\angle GAH=\angle FCH$. Now applying sine rule in $$\Delta GAH \implies \frac{GH}{\sin \angle GAH}=\frac{GA}{\sin \angle GHA}$$. But observe that $$\frac{HC}{\sin \angle HFC}=\frac{HF}{\sin \angle FCH}=\frac{GH}{\sin \angle GAH}=\frac{GA}{\sin \angle GHA} \implies \angle GHA=\angle HFC \implies \Delta GAH \cong \Delta HCF$$. Done $\blacksquare$.