Geometry Problems, Online Education

Problem 146. Varignon's Theorem: Quadrilateral, Midpoints, Parallelogram, Area, Perimeter.

In the figure below, ABCD is a quadrilateral of area S. E, F, G, and H are the midpoints of the sides. S1, S2, S3, and S4 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) S1+ S3 = S2 + S4 = S / 4, (4) the area of the Varignon parallelogram is half that of ABCD.

Varignon theorem, Quadrilateral Area, Midpoints

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