Quick proof that the quadrilateral connecting
the midpoints of a rhombus (HEFG below) is a rectangle
Extend GF to intersect the extension of AC at point M.
It is easy to show that triangle GFD is congruent to triangle
Angle ECF and FCM are supplementary. And each of those angles
is equal to the sum of the opposite interior angles. So the adjacent angles
MFC and CFE are equal in sum to half the supplementary angles ECF and FCM.
Therefore, their sum is 90 degrees and their supplement,
EFC is also 90 degrees. In the same way, it can be shown that all the angles
of quadrilateral HEFG are right.
If you have the Geometer's Sketchpad program, you can
view the Sketchpad