Parallel Lines (Thomsen's Hexagon)

The dynamic geometry activities below are from 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.

Parallel Lines (Thomsen's Hexagon)

Notes
1) 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).
2) To construct parallel lines as directed in the accompanying worksheet, use the 'Parallel' tool on the left.
3) To explore what happens in the case of a pentagon as directed in the worksheet, click on the 'Link to pentagon' button.
4) In 3), click on the 'Sequence 9 Actions' button to construct 9 parallel lines.
5) 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?
6) 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.
7) Further explore the perimeter of the formed Thomsen's hexagon at: Perimeter inscribed parallel-hexagon.
8) Further explore the area of the formed Thomsen's 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.
9) For an interesting variation of Thomsen's hexagon, go here: Merry Go Round the Triangle
10) 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.

References
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)
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.