Angle AEF+ angle CFE= 180 degrees.

Eg and fg are divided equally to ∠ AEN and ∠ CFM respectively.

So angle GEF= angle AEF/2, angle GFE= angle CFE/2.

Therefore, angle GEF+ angle GFE=90 degrees.

So the angle EGF=90 degrees.

In addition, the goal of the question is wrong (n is marked as m). This method is a very basic method, which can be used to prove the congruence, similarity and tangency judgment of right triangle and circle.