Challenge Level

Draw any convex quadrilateral and find the midpoints of the four edges.

Join the midpoints to form a new quadrilateral.

Try it a few times starting with different convex quadrilaterals.

What do you notice about the new quadrilaterals formed by joining the midpoints?

Does this always happen?

Can you find a counter-example?

Can you make a conjecture?

You may wish to use the GeoGebra interactivity to explore.

Can you prove your conjecture?

Is the area of PQRS always the same fraction of the area of ABCD?

Can you explain why?

What if ABCD is a concave quadrilateral?