Mathematics Experiment #10

Order Out of Chaos?

Pick any 4 points on the coordinate plane. Connect them to form a quadrilateral. Using construction techniques, find the midpoints of each of the sides. Connect these midpoints. Do this at least 10 different times with different choices for the original 4 points of the quadrilateral. Is there a pattern? Is there a theorem for this? Can you prove it?

Is this true in other cases? Try at least twenty different random configurations of lines and points.