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Vectors - Prove the following relation about the centroid

  1. Nov 21, 2008 #1
    1. The problem statement, all variables and given/known data
    http://img147.imageshack.us/img147/733/vectorsba4.png
    Given is the triangle OAB and a variable point P. G is the centroid. Prove that:
    PA2+PB2+PO2=GA2+GB2+GO2+3 (PG2)


    3. The attempt at a solution
    I treat O as the origin.
    Vectors are denoted in bold.
    Position vectors of A, B, P are a,b and p respectively.
    G=a+b/3
    How on earth am I going to prove that:
    9(|p-a|2+|p-b|2+|p|2)=|b-2a|2+|a-2b|2+|a+b|2+3|a+b-3p|2

    I mean is there any criteria to prove this equation in modulus of vectors when I dont know the angles between any of them?
     
  2. jcsd
  3. Nov 21, 2008 #2

    tiny-tim

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    Hi ritwik06! :smile:

    No, it'll be easier ('cos it's more symmetrical) if you treat P as the origin, and use g = (1/3)(a + b + c) :wink:

    (when the answer is symmetrical, always try to keep the proof symmetrical! :wink:)
     
  4. Nov 21, 2008 #3
    Re: Vectors:

    Thanks a lot tim.
    I have done exactly that. And I get:
    a2+b2+c2=1/9(|a+b-2c|2+|b+c-2a|2+|a+c-2b|2+3|a+b+c|2)

    But the fact still remains that I dont know many angle such as th one made by b+c-2a.?? I think it is 0.
     
  5. Nov 21, 2008 #4

    tiny-tim

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    No … start with the complicated part of the equation, and try to simplify it, not the other way round!

    In other words, start with (g - a)2 + (g - b)2 + (g - c)2 :smile:
     
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