If the lines p_ 1 x+q_ 1 y=1, p_ 2 x+q_ 2 y=1 and p_ 3 x+ q_ 3 y=1...

The equations of the lines are $$\begin{array}{l}\mathrm{p}_{1} \mathrm{x}+\mathrm{q}_{1} \mathrm{y}-1=0 \\ \mathrm{p}_{2} \mathrm{x}+\mathrm{q}_{2} \mathrm{y}-1=0 \end{array}$$and $\mathrm{p}_{3} \mathrm{x}+\mathrm{q}_{3} \mathrm{y}-1=0$ As they are concurrent, \left|$\begin{array}{lll} \mathrm{p}_{1}& \mathrm{q}_{1}& -1 \\ \mathrm{p}_{2}& \mathrm{q}_{2}& -1 \\ \mathrm{p}_{3}& \mathrm{q}_{3}& -1 \end{array}$\right|=0 \Rightarrow\left|$$\begin{array}{lll} \mathrm{p}_{1}& \mathrm{q}_{1}& 1 \\ \mathrm{p}_{2}& \mathrm{q}_{2}& 1 \\ \mathrm{p}_{3}& \mathrm{q}_{3}& 1 \end{array}$$\right|=0 This is also the condition for the points $\left(\mathrm{p}_{1}, \mathrm{q}_{1}\right),\left(\mathrm{p}_{2}, \mathrm{q}_{2}\right)$ and \left(\mathrm{p}_{3}, \mathrm{q}_{3}\right)$ to be collinear.