No, it is incorrect. The geometrical way is quite hard to find, but you have to recognize some patterns.
Doing angle addition in this problem is of no use, to spare you some torture.
In the end, I gave up doing this problem, and used the cheap way (Law of Sines, Law of Cosines) to find the answer. However, here are some hints on how to approach the problem (after I found out the geometrical solution):
- Draw a line parallel to $BC$ which intersects $E$ ($EG$)
- Draw a line from $G$ to $C$ ($GC$), and label the intersection it has with $BE$ point $H$.
- Connect $A$ to $H$ (should automatically come to you)
Draw this out in a BIG image, and if you stare at one part of the image, everything should fall in place. Or, maybe you'll have to find several critical angles.
Hint: There are many equilateral triangles (as well as some isosceles triangles).