Thomsen’s Figure

Thomsen’s figure, or Thomsen’s theorem, illustrates the following process: For any triangle ABC, pick a point P₁ on BC opposite A. From P₁ draw a line segment parallel to AB intersecting AC at point P₂; then from there a segment parallel to BC intersecting AB at P₃; continuing through the figure. Thomsen’s theorem shows that the final segment from P₆ parallel to AC meets at the original point P₁.

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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