Vectors - Prove the following relation about the centroid

[ritwik06]

Homework Statement

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)

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 don't know the angles between any of them?

 

[tiny-tim]

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!

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! )

 

[ritwik06]

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! )

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 don't know many angle such as th one made by b+c-2a.?? I think it is 0.

 

[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