The lines whose direction cosines are given by the relations al + bm + cn = 0 and nm + nl + lm = 0 are

$$\begin{gathered} \mathrm{al}+\mathrm{bm}+\mathrm{cn}=0 \\ \Rightarrow \mathrm{l}=-\frac{(\mathrm{bm}+\mathrm{cn})}{\mathrm{a}} \\ \mathrm{lm}+\mathrm{mn}+\mathrm{nl}=0 \\ -\frac{(\mathrm{bm}+\mathrm{cn})}{\mathrm{a}}\mathrm{m}+\mathrm{mn}-\mathrm{n}\left(\frac{\mathrm{bm}+\mathrm{cn}}{\mathrm{a}}\right)=0 \\ -{\mathrm{bm}}^2-\mathrm{cmn}+\mathrm{amn}-\mathrm{bmn}-{\mathrm{cn}}^2=0 \\ \mathrm{b}{\left(\frac{\mathrm{m}}{\mathrm{n}}\right)}^2-(\mathrm{a}-\mathrm{b}-\mathrm{c})\frac{\mathrm{m}}{\mathrm{n}}+\mathrm{c}=0 \end{gathered}$$

$\frac{{\mathrm{m}}_1}{{\mathrm{n}}_1}, \frac{{\mathrm{m}}_2}{{\mathrm{n}}_2}$  are roots of eqn

$$\begin{gathered} \frac{{\mathrm{m}}_1}{{\mathrm{n}}_1}·\frac{{\mathrm{m}}_2}{{\mathrm{n}}_2}=\frac{\mathrm{c}}{\mathrm{b}} \\ \mathrm{m}=\frac{-(\mathrm{al}+\mathrm{cn})}{\mathrm{b}} \\ \mathrm{lm}+\mathrm{mn}+\mathrm{nl}=0 \\ \frac{-\mathrm{l}(\mathrm{al}+\mathrm{cn})}{\mathrm{b}}+\ln -\frac{(\mathrm{al}+\mathrm{cn})}{\mathrm{b}}\mathrm{n}=0 \\ {\mathrm{al}}^2+\mathrm{cln}-\mathrm{bln}+\mathrm{aln}+{\mathrm{cn}}^2=0 \\ \mathrm{a}{\left(\frac{\mathrm{l}}{\mathrm{n}}\right)}^2+(\mathrm{a}-\mathrm{b}+\mathrm{c})\frac{\mathrm{l}}{\mathrm{n}}+\mathrm{c}=0 \end{gathered}$$

$\frac{{\mathrm{l}}_1}{{\mathrm{n}}_1}, \frac{{\mathrm{l}}_2}{{\mathrm{n}}_2}$ are roots of eqn.

$\begin{array}{rcl}\frac{{\mathrm{l}}_1 {\mathrm{l}}_2}{{\mathrm{n}}_1 {\mathrm{n}}_2}& =& \frac{\mathrm{c}}{\mathrm{a}}\\ \frac{{\mathrm{l}}_1 {\mathrm{l}}_2}{{\mathrm{n}}_1 {\mathrm{n}}_2}+\frac{{\mathrm{m}}_1 {\mathrm{m}}_2}{{\mathrm{n}}_1 {\mathrm{n}}_2}+1& =& 0\\ {\mathrm{l}}_1 {\mathrm{l}}_2+{\mathrm{m}}_1 {\mathrm{m}}_2+{\mathrm{n}}_1 {\mathrm{n}}_2& =& 0\end{array}$

lines are ${\perp }^{\mathrm{er}}$

$\begin{array}{rcl}\frac{\mathrm{c}}{\mathrm{a}}+\frac{\mathrm{c}}{\mathrm{b}}+1& =& 0\\ & \Rightarrow & \frac{1}{\mathrm{a}}+\frac{1}{\mathrm{b}}+\frac{1}{\mathrm{c}}=0\end{array}$