6. The solution set of the inequality $\frac{1}{x^{2}-2 x+2}+\lg \frac{1-x^{2}+2 x}{1+x^{2}-2 x}<\frac{1}{2}$ is
Solution: $(1-\sqrt{2}, 0) \cup(2,1+\sqrt{2})$
Let $x^{2}-2 x=t$, we get $\frac{1}{t+2}+\lg \frac{1-t}{1+t}<\frac{1}{2} \cdots \cdots$ (1)
Let $f(t)=\frac{1}{t+2}+\lg \frac{1-t}{1+t}$, we know $f(t)<f(0)=\frac{1}{2}$
The domain of $f(t)$ is $(-1,1)$, and it is monotonically decreasing on this interval, so the solution to (1) is $0<t<1$. Therefore, $0<x^{2}-2 x<1$, and the solution to this inequality is $(1-\sqrt{2}, 0) \cup(2,1+\sqrt{2})$