Question

if the centroid of a triangle formed by the points (a,b) , (b,c) and (c,a) is at the origin find the value of a cube + b cube+ c cube

Answer 1

Coordinates of the Centroid of a triangle = (x1+x2+x3)/3 and (y1+y2+y3)/3

Here x = (a+b+c)/3 , y = (b+c+a)/3.

But in this case the centroid is at the origin that means x = 0 and y=0 or (a+b+c)/3 = 0

so a+b+c = 0.

We know that a3 + b3 +c3 -3abc = (a+b+c)(a2+b2+c2 –ab-bc-ac).

Let us put the value of a+b+c in the above formula.

a3 + b3 +c3 -3abc = 0*(a2+b2+c2 –ab-bc-ac)

or a3 + b3 +c3 -3abc = 0

or a3 + b3 +c3 = 3abc

Answer 2

the value is 3abc.........