# 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 prove this formula we have the given equations of straight lines:

To solve the 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 that lines (i) and (ii) intersect at a point

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

This can be written in determinant form:

This is the condition of concurrency of three straight lines.