Manipulate the equations to get a common denominator: $a=\frac{b+c}{x-2} \Longrightarrow x-2=$ $\frac{b+c}{a} \Longrightarrow x-1=\frac{a+b+c}{a} \Longrightarrow \frac{1}{x-1}=\frac{a}{a+b+c}$; similarly, $\frac{1}{y-1}=\frac{b}{a+b+c}$ and $\frac{1}{z-1}=\frac{c}{a+b+c}$. Thus $\frac{1}{x-1}+\frac{1}{y-1}+\frac{1}{z-1} =1 \Rightarrow (y-1)(z-1)+(x-1)(z-1)+(x-1)(y-1) =(x-1)(y-1)(z-1) \Rightarrow x y+y z+z x-2(x+y+z)+3 =x y z-(x y+y z+z x)+(x+y+z)-1 \Rightarrow x y z-2(x y+y z+z x)+3(x+y+z)-4 =0 \Rightarrow x y z-2(67)+3(2010)-4 =0 \Rightarrow x y z =-5892