In the figure below, ABCD is a
quadrilateral of area S. E, F, G, and H are the midpoints of the
sides. S_{1}, S_{2}, S_{3}, and S_{4} are the areas of triangles AEH,
BEF, CFG, and DGH respectively. Prove that: (1) EFGH is a
parallelogram, called Varignon parallelogram, (2) the
perimeter of the Varignon parallelogram is equal to the sum of
diagonals of ABCD, (3) S_{1}+ S_{3} = S_{2}
+ S_{4} = S / 4,
(4) the area of the Varignon parallelogram is half
that of ABCD.
