Thomsen’s figure, or Thomsen’s theorem, illustrates the following process: For any triangle ABC, pick a point P_{1} on BC opposite A. From P_{1} draw a line segment parallel to AB intersecting AC at point P_{2}; then from there a segment parallel to BC intersecting AB at P_{3}; continuing through the figure. Thomsen’s theorem shows that the final segment from P_{6} parallel to AC meets at the original point P_{1}.

You might enjoy this interactive Wolfram Demonstrations Project version.

Should this work for any convex polygon with an odd number of sides? The arrows here aren’t parallel to the opposite sides, so not the right process.

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 MathWorld 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