Today I was reading some math book when the author mentions Varignon’s theorem, and gives a proof. The proof was not very long, but it was somewhat confusing. On Wikipedia, several more short proofs were given, but they were all more confusing than need be.
I remembered seeing the theorem proven using vector geometry before, but I couldn’t find the text (nor any other page / book that proves it this way) –
[image shamelessly taken from Wikipedia]
Varignon’s theorem states that in any quadrilateral, if we join the midpoints of the sides, then we get a parallelogram.
In the diagram, it suffices to prove that vector HG is equal to vector EF — vectors must have both the same orientation and length to be equal. This works since any method that proves HG = EF can also prove HE = GF. The proof goes as follows –
And we’re done. (the last step is due to symmetry of HG and EF)