Parallel Lines (Thomsen's Hexagon)

The dynamic geometry activities below are 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: Parallel Lines Worksheet & Teacher Notes.

Optional Prerequisite: Knowledge of the proportional intercept theorem (i.e. a line drawn parallel to one side of a triangle divides the other two sides in the same proportion).

Focus Questions
What do you think will happen if we start from an arbitrary point D as shown below on AB and draw a parallel line to AC until the line hits BC at E? Then from point E draw a line parallel to AB until the line hits AC. What do you think will happen if we continue this process of drawing parallel lines to the sides of the triangle in this way? Will it continue forever? Will it ever return to the starting point D? If you think so, how many times do you think it might need to go round to do that? Does the starting point matter or do we get different results for different starting points?

Before proceeding with the activity below, take a guess what you think will happen!

Parallel Lines (Thomsen's Hexagon)

Notes
1a) Click the three green buttons on the left above to construct the first three parallel lines.
1b) To continue constructing parallel lines as directed in the accompanying worksheet, use the 'Parallel' tool on the left.
2) What do you notice eventually? Drag D to check your observations.
3a) How sure are you that the last parallel line you constructed passes exactly through D?
3b) Does the result still hold if you drag D outside the triangle on the extension of AB? Explore.
4) To explore what happens in the case of a pentagon as directed in the worksheet, click on the 'Link to pentagon' button.
5) In 4), click on the 'Sequence 9 Actions' button to construct 9 parallel lines.
6) Again use the 'Parallel' tool on the left, to construct a line through O parallel to BE. What do you notice? Can you prove your observation?
7) Note that the guided proof in the accompanying worksheet in Step 7 only provides empirical verification that a line parallel to a side of a triangle divides (intersects) the other two sides in equal ratios. For a proof of that result go to: Reasoning Backwards: Parallel Lines.

Explore More
8) Further explore the perimeter of the formed Thomsen's hexagon at: Perimeter inscribed parallel-hexagon.
9) Further explore the area of the formed hexagon for a triangle as well as that of the formed parallelogram in the analogous case for a quadrilateral at: Area ratios of some parallel-polygons inscribed in quadrilaterals and triangles.

Further Generalization & Other Connections
10) Can you generalize your results in 1) to 9) to higher polygons? Compare your answer with De Villiers (1994, pp. 76-88).
11) For an interesting variation of this hexagon, go here: Merry Go Round the Triangle
12) Here also is an interesting general connection of Thomsen's hexagon with the so-called Tucker circle, found by Đào Thanh Oai in 2015: A generalization of Thomsen theorem and Tucker circle.

Brief Historical Background
The main purpose of this activity is for students to discover and explain why (prove that) a closed hexagon is formed if we continue drawing parallel lines to the sides of a triangle. This surprising result is known as Thomsen's theorem and named after a German mathematician, Gerhard Thomsen (23 June 1899 – 4 January 1934).
'.....Personally I first learnt of the surprising result about the closure of Thomsen's hexagon from a talk by Shmuel Avital (Technion, Israel) in the late 70's or early 80's. This result is also mentioned in Avital & Barbeau (1991, p. 5). I also recently found the result mentioned in Coxeter (1961), where he mentions that apparently this crossed hexagon is called Thomsen's figure in Nev R. Mind (1953), and named after a German mathematician, Gerhard Thomsen (23 June 1899 – 4 January 1934).

References
Avital, S. & Barbeau, E. (1991). Intuitively Misconceived Solutions to Problems. For the learning of mathematics, Vol 3, Num 3, pp. 2-8.
Coxeter, H. S. M. (1961/1969). Introduction to Geometry. John Wiley & Sons, Inc., USA, pp. 198-199.
De Villiers, M. (1994, 1996, 2009). Generalizing Varignon's Theorem, pp. 76-88 from Some Adventures in Euclidean Geometry (free to download). Dynamic Mathematics Learning.
De Villiers, M. (1999, 2003, 2012). Rethinking Proof with Geometer's Sketchpad (free to download). Key Curriculum Press.
De Villiers, M. (1999). A Sketchpad discovery involving areas of inscribed polygons. Mathematics in School, 28(2), March, 18-21.

Other Rethinking Proof Activities
Other Rethinking Proof Activities

External Links
Thomsen's theorem (Wikipedia)
Thomsen's Figure (Wolfram MathWorld)
Aimssec Lesson Activities (African Institute for Mathematical Sciences Schools Enrichment Centre)
UCT Mathematics Competition Training Material
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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Created by Michael de Villiers with WebSketchpad, 4 July 2025; updated 6 July 2025; 27 July 2025; 13 Jan 2026; 1 April 2026.