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

\left| {\begin{array}{*{20}{c}} {{a_1}}&{{b_1}}&{{c_1}} \\ {{a_2}}&{{b_2}}&{{c_2}} \\ {{a_3}}&{{b_3}}&{{c_3}} \end{array}} \right| = 0


Where {a_1}{b_2} - {a_2}{b_1} \ne 0.

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

\begin{gathered} {a_1}x + {b_1}y + {c_1} = 0\,\,\,\,{\text{ - - - }}\left( {\text{i}} \right) \\ {a_2}x + {b_2}y + {c_2} = 0\,\,\,\,{\text{ - - - }}\left( {{\text{ii}}} \right) \\ {a_3}x + {b_3}y + {c_3} = 0\,\,\,\,{\text{ - - - }}\left( {{\text{iii}}} \right) \\ \end{gathered}

To solve the above equations we use the method of simultaneous equations.

Multiplying equation (i) by {b_2}, we have

{a_1}{b_2}x + {b_1}{b_2}y + {b_2}{c_1} = 0\,\,\,\,{\text{ - - - }}\left( {{\text{iii}}} \right)

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

{a_2}{b_1}x + {b_1}{b_2}y + {b_1}{c_2} = 0\,\,\,\,{\text{ - - - }}\left( {{\text{iv}}} \right)

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

 \Rightarrow x = \frac{{{b_1}{c_2} - {b_2}{c_1}}}{{{a_1}{b_2} - {a_2}{b_1}}}

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

{a_1}{a_2}x + {b_1}{a_2}y + {b_2}{a_1} = 0\,\,\,\,{\text{ - - - }}\left( {\text{v}} \right)

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

{a_2}{a_1}x + {a_1}{b_2}y + {a_1}{c_2} = 0\,\,\,\,{\text{ - - - }}\left( {{\text{vi}}} \right)

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

 \Rightarrow y = \frac{{{a_2}{c_1} - {a_1}{c_2}}}{{{a_1}{b_2} - {a_2}{b_1}}}

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

A\left( {\frac{{{b_1}{c_2} - {b_2}{c_1}}}{{{a_1}{b_2} - {a_2}{b_1}}},\frac{{{a_2}{c_1} - {a_1}{c_2}}}{{{a_1}{b_2} - {a_2}{b_1}}}} \right)

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

\begin{gathered} {a_3}\left( {\frac{{{b_1}{c_2} - {b_2}{c_1}}}{{{a_1}{b_2} - {a_2}{b_1}}}} \right) + {b_3}\left( {\frac{{{a_2}{c_1} - {a_1}{c_2}}}{{{a_1}{b_2} - {a_2}{b_1}}}} \right) + {c_3} = 0 \\ \Rightarrow {a_3}\left( {{b_1}{c_2} - {b_2}{c_1}} \right) - {b_3}\left( {{a_1}{c_2} - {a_2}{c_1}} \right) + {c_3}\left( {{a_1}{b_2} - {a_2}{b_1}} \right) = 0 \\ \end{gathered}

This can be written in determinant form:

\left| {\begin{array}{*{20}{c}} {{a_1}}&{{b_1}}&{{c_1}} \\ {{a_2}}&{{b_2}}&{{c_2}} \\ {{a_3}}&{{b_3}}&{{c_3}} \end{array}} \right| = 0

This is the condition of concurrency of three straight lines.