In the interactive Web Sketchpad model below (and here on its own page), *ABCD* is an arbitrary quadrilateral whose midpoints form quadrilateral *EFGH*. Drag any vertex of *ABCD*. What do you notice about *EFGH*?

The midpoint quadrilateral theorem, attributed to the French mathematician Pierre Varignon, is relatively new in the canon of geometry theorems, dating to 1731. Mathematics educator Chris Pritchard says the theorem “epitomises simplicity, economy, and elegance….And even today, in an idle moment, I find myself doodling…perhaps still seeking a quadrilateral for which Varignon’s theorem doesn’t hold.”

Varignon theorem is also a poster child for the benefits of dynamic geometry software, appearing regularly in articles that introduce readers to the pedagogical opportunities afforded by the software. Below are four commonly cited ways in which Varignon’s theorem illuminates the power of dynamic geometry:

- The theorem is dramatic and unexpected: Who would believe that a random quadrilateral
*ABCD*has the DNA of a parallelogram within it, ready to be uncovered by connecting its midpoints? Dynamic geometry amplifies the discovery and the ‘wow’ factor, showing not just one parallelogram but an entire collection of them as you drag the construction. - The theorem nicely illustrates the ability of dynamic geometry software to explore “monster” cases that normally do not receive attention. Using the software, you can drag a vertex of
*ABCD*so that the quadrilateral becomes pretzel-like in appearance and still*EFGH*is a parallelogram. - The theorem demonstrates the role of dynamic geometry in establishing our conviction that
*EFGH*is a parallelogram. Had you viewed a single static image of the construction, you would have little reason to believe that*EFGH*was always a parallelogram. But the data you obtain from dragging the construction leaves little doubt about what you’re observing. With this firm belief about*EFGH’*s properties in place, proof becomes a vehicle not for assuaging our doubts but rather for helping you to understand*why*the quadrilateral is a parallelogram. - The proof of the theorem can be scaffolded with the help of the software. Ask students to construct the two diagonals
*AC*and*BD*of*ABCD*and see what they observe and can figure out. Students might*AC, EF,*and*GH,*and then drag the construction, noticing that*AC*is always twice the length of*EF*and*GH.*The combination of the diagonal and the measurements helps them to recognize the triangle midsegment theorem applied to Δ*ABC*and Δ*ADC*.

- The theorem is dramatic and unexpected: Who would believe that a random quadrilateral

I think items 1–3 above showcase dynamic geometry at its best, but I’m less enthusiastic about dynamic geometry software’s role in the proof of Varignon’s theorem in item 4. Telling students to construct the diagonals of the parallelogram feels like we’re depriving them of the *aha!* moment of realizing that the proof is nothing more than several applications of the triangle midsegment theorem. Below are two ways that we might make better use of dynamic geometry to motivate a proof of Varignon’s theorem. Each focuses on a degenerate case of the construction, turning quadrilateral *ABCD* into a triangle.

##### Ask students to drag a vertex of *ABCD* so that two sides of the parallelogram overlap

While exploring this task, students discover that the overlapping occurs when two opposite vertices of *ABCD* coincide (points *B* and *D* in the picture below). In fact, not only do the sides of the presumed parallelogram overlap, but its remaining two sides disappear entirely. The image below is certainly suggestive of the triangle midsegment theorem—it’s only missing segment *AC*. To gather more visual data, students can drag point *C* to view a collection of triangles *ADC* (with missing side *AC*), all displaying their midsegment.

##### Ask students to drag a vertex of *ABCD* so that the vertex is collinear with its two adjacent vertices

The picture below shows one such location of point *B*. By construction, *EB* is always half the length of *AB* and *FB* is always half the length of *CB *regardless of point *B’*s location*.* Segment *EF,* which is composed of *EB* and *FB,* is thus half the length of *AC*. If we assume, based on our observations of dragging point *B,* that *GH* is equal to *EF,* then *GH* is equal to half the length *AC*. This statement rings a bell—it’s the triangle midsegment theorem. By assuming that *EF* = *GH,* we’ve found a result that helps us to backtrack and explain why the two lengths are equal.

And as a bonus, the picture below suggests another theorem: Start with a triangle *ACD* and construct midpoints *G* and *H* on two of its sides. On the third side, *AC,* construct midpoint *B* and then construct midpoints *E* and *F* of *AB* and *CB, *respectively. Connect points* E, F, G,* and *H*. They form a parallelogram. You can view this model on the second page of the websketch at the top of the page by clicking the arrow in its lower-right corner.

##### Varignon’s Converse

Finally, the Web Sketchpad model below (and here on its own page) shows a fascinating variation of Varignon’s theorem described by Celia Hoyles in *The Changing Shape of Geometry*. We start with a random quadrilateral *ABCD* and a point *E*. Construct a segment *EF* passing through vertex *B* so that *EB* = *BF*. Then construct *FG,* passing through vertex *C,* so that *FC* = *CG*. Continue this construction method with *GH* and *HI.* Now, drag point *E*. Notice that vector *IE* maintains its length and direction. Can you prove why?

Hoyles writes, “…when I noticed the invariant vector and could explain it, my perspective on the original theorem completely changed. I now see Varignon as a particular case of a more general theorem, which can be stated informally as follows: every quadrilateral has associated with it a fixed vector *IE,* which just happens to be zero when the original quadrilateral is a parallelogram.”

If you rub out the lines AD and CD, and connect A to C and E to F then moving point B around gives you more of a clue.