Thomsen's theorem

Thomsen's theorem, ${\displaystyle P_{7}=P_{1}}$

The set of Thomsen (by Gerhard Thomsen ) is a statement in the elementary geometry that states that a certain over parallel line segments in the triangle constructed polyline always ends at its starting point.

For any triangle ABC with a point P ₁ on the side BC , one constructs the following parallels and intersections . The parallel to AC through P ₁ intersects AB in P ₂ . The parallel to BC through P ₂ intersects AC in P ₃ and the parallel to AB through P ₃ intersects BC in P ₄ . The parallel to AC through P ₄ intersects AB in P ₅ and the parallel to BC through P ₅ intersects AC in P ₆ . Finally, consider the parallel to AB through P ₆ , which BC intersects in P ₇ . Thomsen's theorem now states that P ₇ and P _{1 are} identical, i.e. the line P ₁ P ₂ P ₃ P ₄ P ₅ P ₆ P ₇ created by intersections with the parallels always ends at its starting point. So you always get a closed route P ₁ P ₂ P ₃ P ₄ P ₅ P ₆ P ₁

Thomsen's theorem is identical to the affine form of Pappos' dual small theorem and, as a Thomsen figure, plays a role in the coordination of axiomatically defined projective planes .

Web links

• Darij Grinberg: Closing theorems in plane geometry (PDF)
• Eric W. Weisstein : Thomsen's Figure . In: MathWorld (English).

Individual evidence

1. ↑ Theorem by Thomsen In: Schülerduden - Mathematik II . Bibliographisches Institut & FA Brockhaus, 2004, ISBN 3-411-04275-3 , pp. 358-359
2. ^ W. Blaschke: Projective Geometry . Springer-Verlag, 2013, ISBN 3-0348-6932-0 , p. 190