Logical Discovery: Varignon Parallelogram Perimeter

"... the end of problem-solving may not be solutions so much as new problems." - C.T Daltry et al (1966:20).

The geometry learning activity below is from the "Proof as Discovery" 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: Logical Discovery: Varignon Parallelogram Perimeter Worksheet & Teacher Notes.

Prerequisites
Before doing this activity, it is recommended that you first complete this Rethinking Proof activity: Kite Midpoints

Logical Discovery
In the preceding Rethinking Proof Activities, you may have discovered geometric properties by first making a construction in Sketchpad and then producing a logical explanation as to why the property must hold true.
In mathematical research, however, experimentation does not always precede logical reasoning. As you will see in this activity, people also discover new geometric properties by logical reasoning first. Only afterward do they follow up with construction and measurement to make sure that false assumptions or conclusions have not been made.

No Computer
(A dynamic geometry sketch is not used in the logical derivation below)

Discovering: Varignon Parallelogram Perimeter
You should have discovered previously in the Kite Midpoints activity that if you connect the midpoints of the sides of any quadrilateral, you get a parallelogram.
This result for a general quadrilateral is known as Varignon's theorem, named after Pierre Varignon, who first provided a logical explanation for it in 1731. Now, without using construction or measurement, work through the following questions using the diagram shown below, where E, F, G and H are the midpoints of the sides of ABCD.

Logical Discovery Varignon Perimeter

1. Write an equation relating the lengths EF and HG to the length AC.

2. Write an equation relating EH and FG to BD.

3. Explain how you found your equations in Questions 1 and 2.

4. Use Questions 1 and 2 to describe the relationship between the perimeter of the inscribed parallelogram EFGH and the diagonals of quadrilateral ABCD.

5. Do you think the relationship you wrote down in Question 4 also holds when ABCD is concave or crossed? On a piece of paper draw a concave example of ABCD and one where ABCD is crossed. Check whether your arguments in 1, 2 and 4 above still apply to both these examples.

Dynamic Sketch
(Use the dynamic geometry sketch below to check the logical derivation above)

Check by Construction
6. Make constructions with appropriate measurements in Sketchpad, or use the dynamic sketch below, to confirm your conclusions from Questions 4 & 5.
(In the sketch below, use the Tools on the left to measure the perimeter of ABCD, the lengths of the diagonals, and calculate their sum with the calculator.
Use the vertical Scroll bar to scroll down to Tools further down in the menu. Also note that you can click on any measurements in the sketch & by holding down, can move them around).
7. Be sure to check the concave and crossed cases for quadrilateral ABCD. Summarize your results. Your summary may be in paper form or electronic form and may include a presentation sketch in Sketchpad or a screengrab of the dynamic sketch on this webpage. You may want to discuss the summary with your partner or group.

Checking Logical Discovery: Varignon Parallelogram Perimeter

Explore More
8. Can you generalize to pentagons, hexagons, etc.?
9. Specifically, what can you conjecture about the perimeters of the polygons formed by the midpoints of the sides of pentagons, hexagons, etc.?
10. Check your conjecture in 9), by clicking on the 'Link to pentagon' and 'Link to hexagon' buttons to navigate to new sketches.
11. Ensure that you also check concave & crossed cases.
12. Challenge: Can you explain why (prove that) your observations in 9, 10 & 11 above are valid?

Notes
a) This is the second activity in Rethinking Proof which focusses on introducing students to the discovery function of logical reasoning and proof. The intention here is to show how we can discover new results by using known or assumed results and applying deductive reasoning.
b) The result that the perimeter of the Varignon parallelogram EFGH is equal to the sum of the diagonals of ABCD has been called Guinto's corollary by the teacher of a student who discovered it in class (Denson, 1989).
c) The Varignon Parallelogram Perimeter result can be further explored & generalized in different ways, for example, go to these activities:
Perimeter of inscribed parallel-hexagon
Parallel Lines (Thomsen's Hexagon) (Rethinking Proof activity)
Area ratios of some parallel-polygons inscribed in quadrilaterals and triangles
d) The Varignon Parallelogram Perimeter result can also be used, in conjuction with the triangle inequality, to prove the upper bound of an inequality related to the perimeter of a quadrilateral and the sum of its diagonals: see Perimeter-Diagonal Inequality.

References
Daltry, C.T. et al. (1966). Beginning with problems. The development of mathematical activity in children: the place of the problem in this development. Gt. Britain: ATM.
Denson, P. (1989). Guinto's corollary. Mathematics Teacher, March, pp. 160-161.
De Villiers, M. (1994, 1996, 2009). Generalizing Varignon's Theorem, pp. 76-88 from Some Adventures in Euclidean Geometry (free to download). Lulu Press: Dynamic Mathematics Learning.
De Villiers, M. (1998). A Sketchpad discovery involving triangles and quadrilaterals. KZN Mathematics Journal, 3(1), 11-18.
De Villiers, M. (1999, 2003, 2012). Rethinking Proof with Geometer's Sketchpad (free to download). Key Curriculum Press.
Oliver, P.N. (2001). Pierre Varignon and the Parallelogram Theorem. Mathematics Teacher, Vol. 94, No. 4, April, pp. 316-319.
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

External Links
Varignon's theorem (Wikipedia)
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, 3 August 2025; updated 27 Oct 2025.