Question

If the centroid of the triangle formed by the points (a,b) , (b,c) and (c,a) is at the origin, then find the value of `a^3+b^3+c^3`.

Answer 1

Centroid of the given triangle will be `((a+b+c)/3,(b+c+a)/3).`
We are given, centroid is at the origin.
`:. (a+b+c)/3 = 0 and (b+c+a)/3 = 0`
`=>a+b+c = 0`
We know,
`a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca)`
Here, `a+b+c = 0`,
`:. a^3+b^3+c^3-3abc = 0`
`=>a^3+b^3+c^3 = 3abc`