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 MFC.
- 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
file.