If three lines whose equations are y = m1x + c1, y = m2x + c2 and y = m3x + c3

It is given that

y = m₁x + c₁ ….. (1)

y = m₂x + c₂ ….. (2)

y = m₃x + c₃ ….. (3)

By subtracting equation (1) from (2) we get

0 = (m₂ – m₁) x + (c₂ – c₁)

(m₁ – m₂) x = c₂ – c₁

So we get

\(x= \dfrac{c_2-c_1}{m_1-m_2}\)

Putting in eqn (i), we get,

\( y=\dfrac {m_1(c_2-c_1)}{m_1-m_2}+c_1\)

\( y= \dfrac{m_1c_2-m_2c_1}{m_1-m_2}\)

Putting it the eqn(iii), we get

\( y= \dfrac{m_1c_2-m_2c_1}{m_1-m_2}\)

= \( y= \dfrac{m_3(c_2-c_1)+m_1c_3-m_2c_3}{m_1-m_2}\)

Rearranging, we get

\( m_1​(c_2​−c_3​)+m_2​(c_3​−c_1​)+m_3​(c_1​−c_2​)=0\)