I wil appretiate any suggestion of how to demostrate

this equation:

if $\displaystyle a+b+c=0$

then $\displaystyle (a^2+b^2+c^2)(a^3+b^3+c^3)/6=(a^5+b^5+c^5)/5$

I could investigate for myself some features and

found some interesting expressions, I list them below:

$\displaystyle (a+b+c)^3=a^3+b^3+c^3+3(a+b)(b+c)(a+c)$................(1).

$\displaystyle =a^3+b^3+c^3+3(a+b+c)(ab+bc+ac)-3abc$.........(2).

$\displaystyle =a^3+b^3+c^3+3a^2(b+c)+3b^2(a+c)+3c^2(a+b)+6abc$....(3).

a mean they are all equal (1)=(2)=(3).

and if $\displaystyle a+b+c=0$ then

(4)......$\displaystyle 3abc=a^3+b^3+c^3$.........see exppretion 2.

(5)......$\displaystyle -2(ab+bc+ac)=a^2+b^2+c^2$.........It is well known that$\displaystyle (a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ac)$.

I hope these ecuations above help.