# Condition for Concurrency of Three Straight Lines

The conditions of concurrency of three lines , and is given by

Where .

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 , we have

Multiplying equation (ii) by , we have

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

Multiplying equation (i) by , we have

Multiplying equation (ii) by , 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 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.