If $a^3+b^3+c^3=3abc$, then prove that $a=b=c$.
I see nothing helpful. Is the identity I am using is right?
I think you need a further condition otherwise
a=-2x
b=x
c=x
satisfies your equation for all x.
If you have the condition that a,b,c > 0 then what you can do is let
$a=x+\delta$
$b=x$
$c=x$
and solve for $\delta$ showing it must be equal to zero.
Then by symmetry of the problem the same result applies to $b$ and $c$.
let us go out in a different way.
Lets see if this criteria is satisfied for 3 cases then it will work.It may not be correct way.
let us consider a=b=c=3
a=b=c=7
a=b=c=5
a=b=c=1
we can see that for 3,5,7 the criteria a3+b3+c3=3abc, will not satisfy for 1 only it will satisfy .
Also for other combinations of a,b,c it will not work.