Consider a triangle ABC with centroid at G and the coordinaes be

 

A(0,0); B(x₁,y₁); C(x₂,y₂)

 

then coordinates of point 'G' the centroid of the triangle is given by

 

x = (0 + x₁ + x₂) /3 = (x₁ + x₂) /3

 

y = (0 + y₁ + y₂) /3 = (y₁ + y₂) /3

 

Now to prove

 

AB²+BC²+CA² = 3(GA²+GB²+GC²)

 

Substitute the coordinates at each sides and simplify to obtain above result.

 

Similarly take O(x,y) to prove second result