# Condition for Concurrency of Three Straight Lines

The conditions of concurrency of three lines ${a_1}x + {b_1}y + {c_1} = 0$, ${a_2}x + {b_2}y + {c_2} = 0$ and ${a_3}x + {b_3}y + {c_3} = 0$ is given by

Where ${a_1}{b_2} - {a_2}{b_1} \ne 0$.
To proof this formula we have the given equations of straight lines

Solving above equations we use the method of simultaneous equations.
Multiplying equation (i) by ${b_2}$, we have

Multiplying equation (ii) by ${b_1}$, we have

Now subtracting (iv) from equation (iii), we get

Multiplying equation (i) by ${a_2}$, we have

Multiplying equation (ii) by ${a_1}$, we have

Now subtracting (vi) from equation (v), we get

This shows the lines (i) and (ii) intersect at a point

If the three lines (i), (ii) and (iii) are concurrent, i.e. three lines intersect at one point, then point $A$ must lie on the line (iii) and must satisfy (iii), so

This can be written in determinant form as given by

This is the condition of concurrency of three straight lines.