Vectors - Prove the following relation about the centroid

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Homework Statement


http://img147.imageshack.us/img147/733/vectorsba4.png [Broken]
Given is the triangle OAB and a variable point P. G is the centroid. Prove that:
PA2+PB2+PO2=GA2+GB2+GO2+3 (PG2)


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?
 
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Answers and Replies

  • #2
tiny-tim
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Given is the triangle OAB and a variable point P. G is the centroid. Prove that:
PA2+PB2+PO2=GA2+GB2+GO2+3 (PG2)

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
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:)
 
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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:)
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.
 
  • #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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