Three lines are said to be concurrent if they pass through a common point, i.e. they meet at a point.
Thus, if three line are concurrent the point of intersection of two lines lies on the third line. Let the three
concurrent lines be

`a_1 x + b_1 y + c_1 = 0` .............`(i)`

`a_2 x + b_2 y + c_2 = 0` ..................`(ii)`

`a_3 x + b_3 y + c_3 = 0` .................`(iii)`

Then the point of intersection of equation (i) and (ii) must lie on the third.

The coordinates of the point of intersection of equation (i) and (ii) are

`((b_1c_2-b_2c_1)/(a_1b_2-a_2b_1) , (c_1a_2-c_2a_1)/(a_1b_2-a_2b_1))`

This point lies on the line (iii). Therefore, we get

` => a_3((b_1c_2-b_2c_1)/(a_1b_2-a_2b_1))+b_3 ((c_1a_2-c_2a_1)/(a_1b_2-a_2b_1))+c_3 = 0`

` => a_3 (b_1c_2-b_2c_1) + b_3 (c_1a_2-c_2a_1) + c_3 (a_1b_2-a_2b_1) = 0`

` => | ( a_1, b_1, c_1), (a_2, b_2, c_2 ), ( a_3, b_3 , c_3)| =0`



This is the required condition of concurrency of three lines.

`text(Alternative Method :)`

Three lines `L_1 = a_1x + b_1y + c_1 = 0;` `L_2 = a_2x + b_2y+c_2 = 0;` `L_3 = a_3x + b_3y + c_3 = 0`

are concurrent if there exist constants `lamda_1,lamda_2,lamda_3` not all zero at the same time so that `lamda_1L_1 + lamda_2L_2 + lamda_3L_3 = 0,`

i.e. , `lamda_1 (a_1x + b_1y + c_1) + lamda_2 (a_2x + b_2y + c_2) + lamda_3 (a_3x + b_3y + c_3) = 0.`