To review the setup, G is the intersection of the angle bisectors of A and B. Our goal here is to prove that CG is actually the angle bisector of C. Therefore, we can't assume that angles GCE and GCD are the same.

D, E, and F are defined so that GD, GE, and GF are perpendicular to the sides. That's how we know that triangles CGD and CGE are both right triangles. Since we can't assume that CG is an angle bisector, I'm not sure of how we would otherwise show that the two triangles are congruent.

Note: Proofs like this (the other "center of a triangle" proofs are often similar too) can be a little confusing because you need to be careful about what you are assuming in the problem. We can't assume that three lines intersect at one point. We do know, however, that two lines intersect at a point. We then create a line using that point and show that it is actually the line we want.