INMO 2000 Problem 3

If $a, b, c, x$ are real numbers such that $a b c \neq 0$ and

$$\frac{x b+(1-x) c}{a}=\frac{x c+(1-x) a}{b}=\frac{x a+(1-x) b}{c}$$

then prove that either $a+b+c=0$ or $a=b=c$.