Vector Proof Centroid Theorem

  1. I am trying to figure out how to prove the 2:1 ratio of a triangle's medians at the centroid using vectors. Example if I had a triangle ABC with midpoints D of BC, E of AC and F of AB. I know G is where the medians intersect. I have seen many proofs and understand the process that proves the addition of the vectors from the centroid to the vertices are zero i.e. GA+GB+GC=0.
    vectorsum_dreieck.GIF
    Does this prove the 2:1 ratio? I cannot find anything explaining how to prove the actual 2:1 ratio. I am not sure if I am missing something or what. Any help or suggestions would be appreciated. Thanks.
     
  2. jcsd
  3. tiny-tim

    tiny-tim 26,053
    Science Advisor
    Homework Helper

    Hi pjallen58! :smile:

    Yes, because, for example, a + 2(d) is 1/3 of the way,

    and that's a + 2(1/2(b + c)), = a + b + c! :smile:
     
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