Prove that the quadrilateral formed by joining the midpoints of a quadrilateral is a parallelogram. What if the original quadrilateral is a rectangle? What if it is a square?

Given ABCD is a quadrilateral. And EFGH be the mid-points of the above quadrilateral.

Then,

We know that,

If the mid-points of adjacent sides of a triangle are joined, then the line joining those mid-points will be parallel to the third side and half of its length.

Therefore, from triangle’s ABD and BCD,

EF//DB and HG//DB

⇒ EF//HG

Similarly, from triangle ACD and triangle ABC,

FG//AC and EH//AC

⇒ FG//EH

Since EF//HG ,

And FG//EH ,

Therefore EFGH forms a parallelogram for any type of quadrilateral taken.

Hence proved.

If original quadrilateral was a rectangle then,

Then the new quadrilateral formed would be a rhombus as shown in figure, due to

Equal sides are generated because of equal opposite lengths of original quadrilateral.

Therefore,

And diagonals are perpendicular to each other. Hence EFGH forms a rhombus.

If original quadrilateral was a square then,

Then the new quadrilateral formed would be a square as shown in figure, due to

Equal and perpendicular adjacent sides are generated because of equal lengths of original quadrilateral. Even new diagonals are equal and perpendicular.

Therefore,

And diagonals are equal and perpendicular to each other. Hence EFGH forms a square.