In the figure below, ABCD is a
quadrilateral of area S. E, F, G, and H are the midpoints of the
sides. If S1, S2, S3, and S4 are the areas of triangles AEH,
BEF, CFG, and DGH respectively, (1) prove that EFGH is a
parallelogram, called Varignon parallelogram, (2) prove that the
perimeter of the Varignon parallelogram is equal to the sum of
diagonals of ABCD, (3) prove that:
,
(4) prove that the area of the Varignon parallelogram is half
that of ABCD.
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