<s>[INST] How can I prove this for $a+b+c=0$ We wish to prove that
$$\left(\frac{a-b}{c}+\ \frac{b-c}{a}+\frac{c-a}{b}\right)\left(\frac{c}{a-b}+\frac{a}{b-c}+\frac{b}{c-a}\right)=9,$$ for $a+b+c=0$ and $$abc \neq 0 , a\neq b , b\neq c, c \neq a.$$
I tried working with only $2$ fractions at a time, switching some of the variables (ex.$a=-b-c$) and then adding them up, and multiplying first but could not go further.
Thank you in advance.
 [/INST] You can expand directly, by noticing that the first parantheses is
$$\frac{ab(a-b) + bc(b-c) + ca(c-a)}{abc} = -\frac{(a-b)(b-c)(c-a)}{abc}$$
So the expression becomes
$$\frac{a^3+b^3+c^3 + 3abc - a^2b - a^2c - b^2a -b^2c - c^2a - c^2b}{abc}$$
Collecting $a^3, -a^2b, - a^2c $ and similarly this becomes
$$\frac{a^2(a - b -c) + b^2(b-c-a) + c^2(c - a -b)}{abc} + 3$$
Now $a-b-c = 2a$ by the condition so this becomes just
$$2\frac{a^3+b^3+c^3}{abc} + 3$$
But there is an identity $a^3+b^3+c^3 - 3abc = (a+b+c)(a^2+b^2+c^2 - ab - bc - ca)$, so we get $a^3+b^3+c^3 = 3abc$.
Thus we get that the initial expression is $9$.
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