$a+b+c=0$. $a,b,c \in \mathbb{R}^{*}$. Prove that $\frac {1}{-a^2+b^2+c^2} + \frac{1}{a^2-b^2+c^2} + \frac{1}{a^2+b^2-c^2} = 0$

Question

The following problem proves how some equations can be solved with very little data.

Let a, b, c be real numbers $a, b, c \in R*$ such that $a+b+c=0$. Prove that

$\frac {1}{-a^2+b^2+c^2} + \frac{1}{a^2-b^2+c^2} + \frac{1}{a^2+b^2-c^2} = 0$

for any a, b, c real numbers.

Do not use anything beyond 8th grade's curriculum and no geometry to solve this.

I have tried brute-forcing it by bringing it to the common denominator. It doesn't work, however, as the expression gets more and more complicated. Neither does writing a in terms of b and c seem to work, as you get sums at the denominators. The last attempt was to try and prove that the 2 expressions, the given one and the ugly one are equivalent.

Answer 1

Hint: Multiplying to a common denominator in the identity gives $$ (a + b + c)(a + b - c)(a - b + c)(a - b - c)=0. $$

Answer 2

Hint :$-a^2+b^2+c^2=-2bc$

Now try putting this in the original expression