If $a\ne b\ne c$, if $ax+by+c=0$, $bx+cy+a=0$,$cx+ay+b=0$  are concurrent then the value of $2^{a^2{b}^{-1}{c}^{-1}}{.2}^{b^2{c}^{-1}{a}^{-1}}{.2}^{c^2{a}^{-1}{b}^{-1}}$

         

 if $ax+by+c=0$, $bx+cy+a=0$,$cx+ay+b=0$  are concurrent

$\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|=0$

$a+b+c=0\Rightarrow a^3+b^3+c^3=3abc$

$2^{a^2{b}^{-1}{c}^{-1}}{.2}^{b^2{c}^{-1}{a}^{-1}}{.2}^{c^2{a}^{-1}{b}^{-1}}$

$=2^{\frac{a^2}{bc}+\frac{b^2}{ca}+\frac{c^2}{ab}}$

$=2^{\frac{a^3+b^3+c^3}{abc}}$

$=2^3=8$