<Tri-2023-2043>, hint

Triangles ADE and ADG are congruent <AGD=<ABD=<B and 

AB=AE and sp <AEG=<B . Hence A,D,E,G are concyclic <GEF =<A/2.

AG/AC=AB/AC=DF/DC =DA/DC. hence AD//GF. <CGF=<CAD=<A/2=<GEF

Hence circumcircle of triangle EFG is tangent to AC.