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Math Help - Vector Proof Centroid Theorem

  1. #1
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    Vector Proof Centroid Theorem

    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.

    Vector Proof Centroid Theorem-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.
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  2. #2
    MHF Contributor red_dog's Avatar
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    \overrightarrow{GB}+\overrightarrow{GC}=-\overrightarrow{GA}=\overrightarrow{AG}
    But, \overrightarrow{GB}+\overrightarrow{GC}=2\overrigh  tarrow{GD}
    Then 2\overrightarrow{GD}=\overrightarrow{AG}\Rightarro  w \left|2\overrightarrow{GD}\right|=\left|\overright  arrow{AG}\right|\Rightarrow 2GD=GA
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  3. #3
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    The best proof of this geometrical fact that I know is found in the classic text by Davis & Snider. If you have access to a reasonably good mathematics library you can find their text.
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  4. #4
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    I knew I was missing something obvious. Thanks a bunch Red Dog.
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