Varignon Parallelogram Area

The dynamic geometry activity below is from the "Proof as Verification" section of my book Rethinking Proof (free to download).

Worksheet & Teacher Notes
Open (and/or download) a guided worksheet and teacher notes to use together with the dynamic sketch below at: Varignon Area Worksheet & Teacher Notes.

Recommended Preceding Activities
While not essential, it is recommended to have already completed the following two Rethinking Proof activities before engaging with the one below:
Kite Midpoints
Area Ratios

Prerequisites: Knowledge of Varignon's theorem, properties of parallelograms, conditions for congruency, area formula for triangle.

Varignon Parallelogram Area

Notes
1) Having done the preceding Area Ratios activity, it is expected that students will be a bit more skeptical here about their dynamic geometry observations and hopefully more motivated to seek additional verification or conviction. The main focus of this activity is therefore on introducing the verification function of proof.
2) Click on the 'Show areas' button to display the areas of quadrilaterals ABCD and EFGH.
3) Use the 'Calculate' tool on the left to determine the ratio of the two areas. Alternatively, click on the 'Show ratio of areas' button.
4) Drag any of of the vertices of ABCD to check your calculation or observation in 3).
5) How sure are you that the ratio remains constant?
6) Formulate a conjecture.
7) Challenge: Can you prove your conjecture in 6)?
For a hint to a proof, click 'Translate EFGH' button. Then follow the hints & directions in the accompanying worksheet (see link at the top).
8) Does this proof also work if ABCD is concave? If not, can you modify the proof so that it works?

Further Exploration
9) What happens if ABCD becomes a crossed quadrilateral? Drag any one of the vertices over another side: what do you notice about the ratio of the areas?
10) Challenge: Can you prove your observation (conjecture) in 9) for a crossed quadrilateral?.
A proof for 10) is provided in the Teacher Notes (see link at the top) or see De Villiers (2014).

References
Coxeter, H.S.M. & Greitzer, S. L. (1967). Geometry Revisited. The Mathematical Association of America, pp. 51-53.
De Villiers, M. (1999, 2003, 2012). Rethinking Proof with Geometer's Sketchpad (free to download). Key Curriculum Press.
De Villiers, M. (2014). Slaying a Geometrical Monster: Finding the Area of a Crossed Quadrilateral. Scottish Mathematical Council Journal, 44 (Dec), 71-74.
De Villiers, M. (2020), The Value of using Signed Quantities in Geometry. Learning & Teaching Mathematics, No. 29, pp. 30-34.
Oliver, P.N. (2001). Consequences of the Varignon Parallelogram Theorem. Mathematics Teacher, Vol. 94, No. 5, May, pp. 406-408.

Other Rethinking Proof Activities
Other Rethinking Proof Activities

Some External Links
Varignon's theorem (Wikipedia)
Varignon Parallelogram (Cut The Knot)
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)

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Michael de Villiers, created with WebSketchpad, 31 August 2025.