Question

Prove the following identities:
$4\sum (b-c)(b+c-2a{)}^3=9\sum (b-c\left)\right(b+c-a{)}^2$.

Answer 1

$\Rightarrow (b-c){(b+c-2a)}^2+(c-a){(c+a-2b)}^2+(a-b){(a+b-2c)}^2$

$=-9(b-c)(c-a)(a-b)$

and $(b-c){(b+c-a)}^2+(c-a){(c+a-b)}^2+(a-b){(a+b-c)}^2$

$=-4(b-c)(c-a)(a-b)$

and, hence the above relation follows,

$4\sum (b-c){(b+c-2a)}^2$

$=9\sum (b-c){(b+c-a)}^2$