It was too easy for you to find that set of triangles, so you didn't get the real solution.

You have proved △ EAG △ EAF before, so EG=EF and AG=AF.

At △ Berg and △CEF

BE=CE，EG=EF，∠EGB=∠EFC=90

So △ beg △ cef (HL), BG = CF

BG+CF=AB+AG+AC-AF=AB+AC

So BG=CF=4.

AF=AC-CF=5-4= 1