coordinate geometry proof

establish that the quadrilateral formed by joining the mid-points of the sides of the quadrilateral ABCD is a parallelogram

is there anything else about the resulting shape that is special?

establish that a parallelogram is always formed for any quadrilateral with corners at

A(0, 0) ,   B(2m , 2n) ,     C(2p , 2q)    and    D(2r , 2s)

by joining the midpoints of the sides

(a version of Varignon's theorem, if you accept that these coordinate pairs are used without any loss of generality)

as a result of doing this procedure how can you obtain:

• a rhombus?
• a square?
• a rectangle?