If G, H, I, and J are the respective midpoints of the sides AB, BC, DE and EF of a parallelo-hexagon ABCDEF (a hexagon with opposite sides equal and parallel), then area ABCDEF = 2 area GHIJ.
If the midpoints of all the sides of a parallelo-hexagon ABCDEF are connected as shown, then area ABCDEF = 4/3 area GHIJKL (see below).
If G, H, and I, are the respective midpoints of the sides AB, CD, and EF of a parallelo-hexagon ABCDEF are connected as shown, then area ABCDEF = 8/3 area GHI (see below).
Some Parallelo-hexagon Area Ratios
Read my papers Some Hexagon Area Ratios: Problem-solving by related example and/or Proof without words: Parallelohexagon-parallelogram Area Ratio
Michael de Villiers, updated 27 May 2011.