Virginia Mathematics Teacher Spring 2017

Is the Midpoint Quadrilateral Really a Parallelogram? Zoltan Kovacs

Dynamic Geometry software systems (DGS) can be useful and challenging tools in the teaching and learning of reasoning. DGS allow the student to formulate certain geometric facts (e.g. as intermediate steps towards establishing the proof of a given statement) by drawing auxiliary diagrams, and, then, getting convinced of the truth or falsity of the conjectured assertion by checking its validity in many instances, after randomly dragging some elements of the figure. This has already raised some concerns: …increased availability in school mathematics instruction of … dynamic geometry systems… raised the concern that such programmes would make the boundaries between conjecturing and proving even less clear for students… [They] allow students to check easily and quickly a very large number of cases, thus helping students “see” mathematical properties more easily and potentially “killing” any need for students to engage in actual proving. (Lin & al., 2012) Indeed, “dragging” is a characteristic feature of DGS and, therefore, the above expressed worries apply to all DGS. A natural question arises as to is really proving necessary in schools if all

geometric facts are obvious and visible when using DGS? Walter Hickey published a list on the 12 most controversial facts in mathematics in Business Insider in March 2013 (Hickey, 2013). Such a list is always subjective, but surprisingly an easy geometric problem is listed there also, namely Varignon's theorem (see Figure 1). Hickey calls this fact “Midpoint parallelograms” and recalls that by drawing a 4 sided polygon and connecting its midpoints you will get a perfect parallelogram every time (see Figure 2).

Figure 2. A collection of midpoint parallelograms in GeoGebra Materials (Kovács, 2015). Having DGS in use, this fact can be visually checked quite easily by sketching an arbitrary planar quadrilateral, and constructing its midpoints and the polygon they describe. (Here we will use GeoGebra, the free DGS which is an interactive maths application including geometry and algebra, intended for learning and teaching mathematics and science from primary school to university level. It is available at www.geogebra.org.) Parallelograms, by definition, are quadrilaterals with two pairs of parallel sides. Parallelism is actually easy to check roughly, but more careful investigation is needed when a precise answer is required: how do you check if two lines are parallel exactly?

Figure 1 . Varignon's theorem as shown in Wikipedia. The midpoints of the sides of an arbitrary quadrilateral form a parallelogram.

Virginia Mathematics Teacher vol. 43, no. 2

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