Hints:
 Menelaus Theorem

 M. Walter's Theorem

Prove that the perimeter of the central hexagonal region DEFGHM determined by trisection of each side of a triangle ABC and connecting the corresponding points A', A", B', B", C', and C" with the opposite vertex is given by 3/20 the sum of AA', AA", BB', BB", CC', and CC". Post your solutions or ideas in the blog.