If the centroid of a triangle is formed by the points (a,b), (b,c) and (c,a) is at the origin; then a^3 + b^3 + c^3 = ??

Question

If the centroid of a triangle is formed by the points (a,b), (b,c) and (c,a) is at the origin i.e. (0,0) then a^3 + b^3 + c^3 = ??

Answer 1

3abc

Centroid x coordinate = (a + b + c)/3 = 0
Therefore a + b + c = 0

Now, a^3 + b^3 + c^3 -3abc = (a + b + c)(a^2 + ..... + c2)
= 0, as (a + b + c) = 0

Therefore a^3 + b^3 + c^3 = 3abc