Challenge Level

Printable dotty paper

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 what happens as you move points A, B, C and D.

Can you prove your conjecture?

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

Can you explain why?

Once you have had a think about this, you might like to take a look at these two different proofs which have been scrambled up. Can you rearrange them back into their original order?

Do these results still hold if ABCD is a concave quadrilateral?

*A concave quadrilateral is one where one angle is greater than $180^{\circ}$, for example you could draw an "arrowhead" shape.*

If you can find a proof which is different to the ones in our proof sorters, then please do let us know by submitting it as a solution.