Varignon's theorem

Referring to the diagram above, triangles ADC and HDG are similar by the side-angle-side criterion, so angles DAC and DHG are equal, making HG parallel to AC.

Varignon's theorem can also be proved as a theorem of affine geometry organized as linear algebra with the linear combinations restricted to coefficients summing to 1, also called affine or barycentric coordinates.

From the first proof, one can see that the sum of the diagonals is equal to the perimeter of the parallelogram formed.

The lengths of the bimedians can also be expressed in terms of two opposite sides and the distance x between the midpoints of the diagonals.

14  [7]: p. 169 For a self-crossing quadrilateral, the Varignon parallelogram can degenerate to four collinear points, forming a line segment traversed twice.

Area( EFGH ) = (1/2)Area( ABCD )
Proof without words of Varignon's theorem:
  1. An arbitrary quadrilateral and its diagonals.
  2. Bases of similar triangles are parallel to the blue diagonal.
  3. Ditto for the red diagonal.
  4. The base pairs form a parallelogram with half the area of the quadrilateral, A q , as the sum of the areas of the four large triangles, A l is 2 A q (each of the two pairs reconstructs the quadrilateral) while that of the small triangles, A s is a quarter of A l (half linear dimensions yields quarter area), and the area of the parallelogram is A q minus A s .