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 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