Vectors - Prove the following relation about the centroid

1. Nov 21, 2008

ritwik06

1. The problem statement, all variables and given/known data
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)

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?

Last edited by a moderator: May 3, 2017
2. Nov 21, 2008

tiny-tim

Hi ritwik06!

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)

(when the answer is symmetrical, always try to keep the proof symmetrical! )

3. Nov 21, 2008

ritwik06

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.

4. Nov 21, 2008

tiny-tim

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