# Thomsen's theorem

**Thomsen's theorem**, named after Gerhard Thomsen, is a theorem in elementary geometry. It shows that a certain path constructed by line segments being parallel to the edges of a triangle always ends up at its starting point.

Consider an arbitrary triangle *ABC* with a point *P*_{1} on its edge *BC*. A sequence of points and parallel lines is constructed as follows. The parallel line to *AC* through *P*_{1} intersects *AB* in *P*_{2} and the parallel line to BC through *P*_{2} intersects AC in *P*_{3}. Continuing in this fashion the parallel line to AB through *P*_{3} intersects BC in *P*_{4} and the parallel line to *AC* through *P*_{4} intersects *AB* in *P*_{5}. Finally the parallel line to *BC* through *P*_{5} intersects AC in *P*_{6} and the parallel line to *AB* through *P*_{6} intersects *BC* in *P*_{7}. Thomsen's theorem now states that *P*_{7} is identical to *P*_{1} and hence the construction always leads to a closed path *P*_{1}*P*_{2}*P*_{3}*P*_{4}*P*_{5}*P*_{6}*P*_{1}

## References

*Satz von Thomsen*In:*Schülerduden – Mathematik II*. Bibliographisches Institut & F. A. Brockhaus, 2004, ISBN 3-411-04275-3, pp. 358–359 (German)

## External links

- Darij Grinberg:
*Schließungssätze in der ebenen Geometrie*(German) - Weisstein, Eric W. "Thomsen's Figure".
*MathWorld*.