Exploration
1) Start with any ΔABC and an arbitrary point D on AB. From D construct a line parallel to AC, to intersect BC at E.
2) Click on the 'blue' buttons on the top left of the sketch below in sequence to construct EF // BA, FG // BC, etc.
Conjecture
1) What do you notice when you draw the line through I parallel to CB?
2) Click on the last 'blue' button below the sketch to show the perimeters of ABC and DEFGHI. What do you notice?
3) Make two conjectures regarding your observations in 1) and 2). Then carefully CHECK your conjectures by dragging D, or dragging any of the vertices A, B or C.
Perimeter of inscribed parallel-hexagon (Thomsen's hexagon)
Explanation (proof)
Can you explain why (prove that) your two conjectures about closure and the perimeters above are true?
Hint: To help you prove that the hexagon closes, have a look at this dynamic sketch of the same construction with an accompanying guided worksheet: Parallel Lines (Thomsen's Hexagon).
Remark: 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 this paper of his, jointly with Ed Barbeau (Canada), in the journal for the learning of mathematics, Vol 3, Num 3, 1991, p. 5: Intuitively Misconceived Solutions to Problems. 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).
Explore more
1) Can you generalize further to the perimeters of polygons inscribed similarly for quadrilaterals, pentagons, etc.? Go to Logical Discovery: Varignon Parallelogram Perimeter (Rethinking Proof activity) to explore more!
2) Explore the relationship between the area of the inscribed hexagon and that of the triangle, as well as the areas of the analogous cases for quadrilaterals, pentagons, etc. Go here for some interactive sketches: areas of some inscribed polygons.
Reference
Coxeter, H. S. M. (1961/1969). Introduction to Geometry. John Wiley & Sons, Inc., USA, pp. 198-199.
Related Links
Parallel Lines (Thomsen's Hexagon) (Rethinking Proof activity)
Logical Discovery: Varignon Parallelogram Perimeter (Rethinking Proof activity)
Varignon Parallelogram Area (Rethinking Proof Activity)
Reasoning Backwards: Parallel Lines (Rethinking Proof activity)
Reasoning Backward: Triangle Midpoints (Rethinking Proof activity)
Constant perimeter triangle formed by tangents to circle
Some Quadrilateral Inequalities involving Sides & Diagonals
SA Mathematics Olympiad 2022, Round 2, Q25
Light Ray in a Triangle (Fagnano's Minimal Path) (Rethinking Proof activity)
External Links
Thomsen's theorem (Wikipedia)
Thomsen's Figure (Wolfram MathWorld)
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 JavaSketchpad 24 February 2016; updated to WebSketchpad 4 July 2025; updated 28 Nov 2025; 30 March 2026.