Thomsen’s figure, or Thomsen’s theorem, illustrates the following process: For any triangle ABC, pick a point A_{1} on BC opposite A. From A_{1} draw a line segment parallel to AB intersecting AC at point B_{1}; then from there a segment parallel to BC intersecting AB at C_{1}; continuing through the figure. Thomsen’s theorem shows that the final segment from C_{2} parallel to AC meets at the original point A_{1}.

You might enjoy the interactive Wolfram Demonstrations Project version.

Is there any reason this wouldn’t work for any polygon with an odd number of sides?/p>

See:

- “Thomsen’s theorem”, Wikipedia
- Pappus’s hexagon theorem (Dual theorem), Wikipedia
- “Thomsen’s Theorem”, Futility Closet
- “Thomsen’s Figure”, MathWorld
- (same as above, except MW description is wrong)

Designed and rendered using Wolfram Mathematica 13.

First draft December 2021 by Robert Dickau.

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www.robertdickau.com/thomsen.html