NO-ABEL 1999 Problem 3

An isosceles triangle $ABC$ with $AB = AC$ and $\angle A = 30^o$ is inscribed in a circle with center $O$ . Point $D$ lies on the shorter arc $AC$ so that $\angle DOC = 30^o$ , and point $G$ lies on the shorter arc $AB$ so that $DG = AC$ and $AG < BG$ . The line $BG$ intersects $AC$ and $AB$ at $E$ and $F$ , respectively. (a) Prove that triangle $AFG$ is equilateral. (b) Find the ratio between the areas of triangles $AFE$ and $ABC$ .