# Thomsen's theorem

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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.

Gerhard Thomsen was a German mathematician, probably best known for his work in various branches of geometry. In geometry, a polygonal chain is a connected series of line segments. More formally, a polygonal chain P is a curve specified by a sequence of points called its vertices. The curve itself consists of the line segments connecting the consecutive vertices. 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 .

Consider an arbitrary triangle ABC with a point P1 on its edge BC. A sequence of points and parallel lines is constructed as follows. The parallel line to AC through P1 intersects AB in P2 and the parallel line to BC through P2 intersects AC in P3. Continuing in this fashion the parallel line to AB through P3 intersects BC in P4 and the parallel line to AC through P4 intersects AB in P5. Finally the parallel line to BC through P5 intersects AC in P6 and the parallel line to AB through P6 intersects BC in P7. Thomsen's theorem now states that P7 is identical to P1 and hence the construction always leads to a closed path P1P2P3P4P5P6P1 In modern mathematics, a point refers usually to an element of some set called a space. In geometry, an intersection is a point, line, or curve common to two or more objects. The simplest case in Euclidean geometry is the intersection of two distinct lines, which either is one point or does not exist if the lines are parallel.

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• Satz von Thomsen In: Schülerduden – Mathematik II. Bibliographisches Institut & F. A. Brockhaus, 2004, ISBN   3-411-04275-3, pp. 358–359 (German) The International Standard Book Number (ISBN) is a numeric commercial book identifier which is intended to be unique. Publishers purchase ISBNs from an affiliate of the International ISBN Agency.