Varignon's Theorem states that the midpoints of any quadrilateral form a parallelogram.
So what is a parallelogram? There are four criteria for a shape to be classified as a parallelogram.
The first criteria is that the opposite angles must be congruent. Consider the following quadrilateral. What can we change to make this a parallelogram?
The second criteria for a parallelogram is that the opposite sides must be the same length. Consider the next quadrilateral. What can we move to make this a parallelogram?
The third criteria for a parallelogram is that the opposite sides are parallel. Consider the following quadrilateral. What can we move to make this a parallelogram? (Hint: two lines are parallel if they have the same slope.)
The fourth criteria for a parallelogram is for the diagonals to bisect each other, or cut each other perfectly in half. Consider this quadrilateral. What can we change to turn this into a parallelogram?
Now that we know what a parallelogram is, we can see that Varignon's Theorem is true. Play around with the following diagram to watch the parallelogram change with the quadrilateral.[br][br]What shape do you get when both diagonals are the same length?[br]What shape do you get when the diagonals are perpendicular to each other?