If three lines whose equations are `y=m_1x+c_1,y=m_2x+c_2`and `y=m_3x+c_3`are concurrent, then show that `m_1(c_2-c_3)+m_2(c_3-c_1)+m_3(c_1-c_2)=0`.

Explanation

Given lines : `m_(1)x-y+c_(1)=0`…….`(1)`
`m_(2)x-y+c_(2)=0`…….`(2)`
`m_(3)x-y+c_(3)=0`…….`(3)`
Subtracting equation `(2)` from equation `(1)`,
`(m_(1)-m_(2))x+c_(1)-c_(2)=0`
`impliesx=-(c_(1)-c_(2))/(m_(1)-m_(2))`………`(4)`
Subtracting equation `(3)` from equation `(2)`,
`(m_(2)-m_(3))x+c_(2)-c_(3)=0`
`implies x=-(c_(2)-c_(3))/(m_(2)-m_(3))`.........`(5)`
`:'` Given lines are concurrent.
`:.` the values of `x` in equations `(4)` and `(5)` will be same.
`:. -(c_(1)-c_(2))/(m_(1)-m_(2))=-(c_(2)-c_(3))/(m_(2)-m_(3))`
`implies (c_(1)-c_(2))/(m_(1)-m_(2))=(c_(2)-c_(3))/(m_(2)-m_(3))`
`implies m_(1)(c_(2)-c_(3))-m_(2)(c_(2)-c_(3))=m_(2)(c_(1)-c_(2))-m_(3)(c_(1)-c_(2))`
`implies m_(1)(c_(2)-c_(3))-m_(2)(c_(2)-c_(3))-m_(2)(c_(1)-c_(2))+m_(3)(c_(1)-c_(2))=0`
`implies m_(1)(c_(2)-c_(3))+m_(2)(-c_(2)+c_(3)-c_(1)+c_(2))+m_(3)(c_(1)-c_(2))=0`
`implies m_(1)(c_(2)-c_(3))+m_(2)(c_(3)-c_(1))+m_(3)(c_(1)-c_(2))=0`