+226 In the diagram below, AB = AC = 1, and arc BC is centered at A with \angle BAC = 60^\circ. Point D is on \overline{AB}, and point E is on arc BC so that BD = DE, and arc BE (centered at D) is tangent to \overline{AC}. Point F is on \overline{DB}, and point G is on arc DE so that DF = FG, and arc DG (centered at F) is tangent to \overline{AE}. Compute the length DF.
