title: Thomsen's theorem
text: # 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 $P_1P_2P_3P_4P_5P_6P_1$
  
  