Assume, for convenience, that GH=1. Then DH=1 and DG=2. Also, DC=1, BC=3/3 because CG=1 and DB=23/3.
Now, △BDG is iscosceles, so when we draw BM to the midpoint of DG, we have a right angle at M.
So △BMG is a right triangle and now we can nail down the desired cosine.