Prove that G is a center of the circle circumscribed on BED.
So we know that we should prove that G lies on the perpendicular bisectors of every side of the BED triangle. For |ED| we can note that $\triangle EDG$ is equilateral so G, as the opposite vertex of |ED|, has to be on the perpendicular bisector.
And... I'm stuck. How can I show it for the other two? There is a ton of angles here but I don't know how to put them to use. Could you tell me what I should take into account while dealing with geometry problems like this? I don't know what to look at and the amount of information on the picture kind of makes me dizzy and everything looks useful and useless at the same time.