$$\frac{a+b+c}{a^{-1}b^{-1}+b^{-1}c^{-1}+c^{-1}a^{-1}}=abc$$
$a\in \mathrm{C}\setminus -\left(b+c\right),0$
$b\neq 0\text{ and }c\neq 0$
$b\in \mathrm{C}\setminus -\left(a+c\right),0$
$c\neq 0\text{ and }a\neq 0$
$a\in \mathrm{R}\setminus -\left(b+c\right),0$
$b\neq 0\text{ and }c\neq 0$
$b\in \mathrm{R}\setminus -\left(a+c\right),0$
$c\neq 0\text{ and }a\neq 0$
