Consider the lines
$y=m_1 x+c_1 \quad \ldots(i)$
$y=m_2 x+c_2 \quad \ldots(ii)$
Subtracting, we get $0=\left(m_1-m_2\right) x+c_1-c_2$ $x=-\frac{c_1-c_2}{m_1-m_2} \quad$ and $\quad y=-\frac{m_1\left(c_1-c_2\right)}{m_1-m_2}+c_1$ $y=\frac{-m_1\left(c_1-c_2\right)+c_1\left(m_1-m_2\right)}{m_1-m_2}$
Put the values of $x$ and $y$ in $y=m_3 x+c_3$
$\begin{aligned}
&\frac{-m_1\left(c_1-c_2\right)+c_1\left(m_1-m_2\right)}{m_1-m_2}=\frac{-m_3\left(c_1-c_2\right)}{m_1-m_2}+c_3 \\
&\Rightarrow m_1\left(-c_1+c_2+c_1-c_3\right)
\end{aligned}$
$+m_2\left(-c_1+c_3\right)+m_3\left(c_1-c_2\right)=0$
i.e. $m_1\left(c_2-c_3\right)+m_2\left(c_3-c_1\right)+m_3\left(c_1-c_2\right)=0$