Please see diagram below
Part 1 Prove that Angle ∟GBE Congruent to ∟GDE
Part 2 Prove that ΔAGC is isosceles

Proof Part 1

In ΔGBE and ΔGED we have

GB Congruent to GD (Given)
GE Congruent to GE (Its common side between two triangles)
∟BGE Congruent to ∟DGE (given)

So by Side Angle Side congruency we have ΔGBE and ΔGED as congruent so by
Corresponding parts of congruent triangles we have

Angle ∟GBE Congruent to ∟GDE

This proves part 1

Part 2
Prove that ΔAGC is isosceles

In ΔGAF and ΔGFC we have

Angle ∟AGF Congruent to ∟FGC (Given)
GF congruent to GF (common side)
Angle ∟GFA Congruent to ∟GFC (90 each)

So by ASA (Angle Side Angle) Congruency we have ΔGAF and ΔGFC as congruent triangles

So as corresponding parts of congruent triangles are equal we have

AG Congruent to GC
SO AG=GC

Hence triangle AGC is isosceles with AG=GC

Hence Proved