In short, the [[centroid]] of the four points ''A'', ''B'', ''C'', ''D'' is the midpoint of each of the two diagonals ''EG'' and ''FH'' of ''EFGH'', showing that the midpoints coincide.
A second proof requires less algebra. By drawing in the diagonals of the quadrilateral, weone noticenotices two triangles are created for each diagonal. And by the [[midpoint theorem]], the segment containing two midpoints of adjacent sides is both parallel and half the respective diagonal. Since two opposite sides are equal and parallel, we have that the quadrilateral must be a parallelogram.
From the second proof, weone can see that the sum of the diagonals is equal to the perimeter of the quadrilateral formed. Also, we can use vectors 1/2 the length of each side to first determine the area of the quadrilateral, and then to find areas of the four triangles divided by each side of the inner parallelogram.
