Midpoints of a quadrilateral

Midpoints of a quadrilateral: By moving the corners of a quadrilateral, even producing a self-intersecting shape, the shape generated by connecting the midpoints of each line is always a parallelogram. Can you see why? This can be proved using similar triangles or vectors, among other ways.
Investigate what happens if the shape is a triangle (eg make 3 points in a line or move one point to overlap another).