I'll just outline the proof.  You can write it out in
two-column form.  The proof is mainly based on this theorem:

The segment joining the midpoints of two sides of a triangle
is parallel to the third side.

Using that theorem,
DE is parallel to BC because D and E are the midpoints of AB and AC 
in triangle ABC

FH is parallel to BC because F and H are the midpoints of GB and GC 
in triangle ABC

Therefore DE is parallel to FH because they are both parallel to BC

Next draw in line segment AG


 
DH is parallel to AG because D and H are the midpoints of AC and GC 
in triangle ACG

EF is parallel to AG because E and F are the midpoints of AB and BG 
in triangle ABG

Therefore DH is parallel to EF because they are both parallel to AG

Therefore DEFH is a parallelogram because both pairs of opposite
sides are parallel.

-----------------------

Since the diagonals of a parallelogram bisect each other,
DG = GF, and we are given that GF = BF since F is the midpoint of BG.

Similarly, GE = GH, and we are given that GH = HC since H is the midpoint of CG. 

Edwin