Question

If three lines whose rquation are $y=m_{1}x+c_1,y=m_{2}x+c_2$ and $y=m_{2}x+c_2$ concurrent, then show that $m_1\left(c_2-c_1\right)+m_2\left(c_3-c_1\right)+m_3\left(c_1-c_2\right)=0$

Answer 1

$y=m_{1}x+c_1\Rightarrow y-m_{1}x-c_1=0y=m_{2}x+c_2\Rightarrow y-m_{2}x-c_2=0y=m_{3}x+c_3\Rightarrow y-m_{3}x-c_3=0$

For concurrent lines

This must be equal to 0 $\Rightarrow$$\left|\begin{array}{ccc}1& -m_1& -c_1\\ 1& -m_2& -c_2\\ 1& -m_3& -c_3\end{array}\right|=0m_1({-c}_3+c_2)-m_2({-c}_3+c_1)+m_3({-c}_2+c_1)=0\Rightarrow m_1(c_2-c_3)+m_2(c_3-c_1)+m_3(c_1-c_2)=0$

Proved