It is hard to believe that this was only our first discussion using Geogebra, the all-powerful program for visualizing geometry problems. In fact we used just a tiny fraction of the tool’s capabilities to explore four-sided figures, or quadrilaterals.

The first thing we did after drawing quadrilaterals was to also draw the midpoints of each side. Here is an example:
We noticed right away that the newly formed “midpoint quadrilateral” is special in that it looks like a parallelogram. This leads immediately to the question: Is this always the case?

In order to answer this question we turned to tiling: covering the plane with disjoint copies of a single quadrilateral. For more fun, check out our handout on quadrilaterals.