Thomsen's theorem, named after

Gerhard Thomsen Gerhard Thomsen (23 June 1899 – 4 January 1934) was a German mathematician, probably best known for his work in various branches of geometry. Life Thomsen was born on 23 June 1899 in Hamburg. His father, Georg Thomsen, was a physician. Thomsen g ...

, is a theorem in elementary geometry. It shows that a certain

path A path is a route for physical travel – see Trail. Path or PATH may also refer to: Physical paths of different types * Bicycle path * Bridle path, used by people on horseback * Course (navigation), the intended path of a vehicle * Desire p ...

constructed by line segments being parallel to the edges of a

triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...

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 In geometry, parallel lines are coplanar straight lines that do not intersect at any point. Parallel planes are planes in the same three-dimensional space that never meet. ''Parallel curves'' are curves that do not touch each other or int ...

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, , pp. 358–359 (German)

External links

*Darij Grinberg
''Schließungssätze in der ebenen Geometrie''
(German) *{{MathWorld, title=Thomsen's Figure, urlname=ThomsensFigure
Thomsen's Figure
at

Wolfram Demonstrations Project The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...

. Theorems about triangles