Dynamic Geometry Follow-Up

Just to show what you can do with dynamic geometry software, here are some illustrations using the free software, Cinderella.

The task is to draw any quadrilateral, ABCD; locate the midpoints of the sides; connect the midpoints of the sides consecutively to obtain quadrilateral EFGH. Drag free points and make a conjecture about EFGH.

Here are three pictures of what happens when you move what is free to move. First, the diagram at the start:

Next is a picture showing what happens when the original quadrilateral is dragged to make it convex:

Finally a shot of what happens when the dragging makes ABCD into something we wouldn't even call a quadrilateral:

What appears to be true about EFGH in all three illustrations? How could you prove it?