Condition for Concurrency of Three Straight Lines

Theorem
Let $3$ straight lines $\LL_1$, $\LL_2$ and $\LL_3$ be embedded in a cartesian plane $\CC$, expressed using the general equations:

Then $\LL_1$, $\LL_2$ and $\LL_3$ are concurrent :
 * $\begin {vmatrix} l_1 & m_1 & n_1 \\ l_2 & m_2 & n_2 \\ l_3 & m_3 & n_3 \end {vmatrix} = 0$

where $\begin {vmatrix} \cdot \end {vmatrix}$ denotes a determinant.

Necessary Condition
Let $\LL_1$, $\LL_2$ and $\LL_3$ be concurrent.

From Equation of Straight Line through Intersection of Two Straight Lines, $\LL_1$, $\LL_2$ and $\LL_3$ are concurrent $\LL_3$ has an equation of the form:


 * $\paren {l_1 x + m_1 y + n_1} - k \paren {l_2 x + m_2 y + n_2} = 0$

That is:
 * $\paren {l_1 - k l_2} x + \paren {m_1 - k m_2} y + \paren {n_1 - k n_2} = 0$

Hence:

Sufficient Condition
Suppose $\begin {vmatrix} l_1 & m_1 & n_1 \\ l_2 & m_2 & n_2 \\ l_3 & m_3 & n_3 \end {vmatrix} = 0$

Then the rows of this matrix are linearly dependent.

That is, there exists $a_1, a_2, a_3 \in \R_{\ne 0}$ such that:

$\LL_3: l_3 x + m_3 y + n_3 = 0$ is equivalent to $- \dfrac {a_3} {a_1} l_3 x - \dfrac {a_3} {a_1} m_3 y - \dfrac {a_3} {a_1} n_3 = 0$,

which, from above, is in turn equivalent to:


 * $\paren {l_1 x + m_1 y + n_1} - k \paren {l_2 x + m_2 y + n_2} = 0$

with $k = -\dfrac {a_2} {a_1}$.

Hence from Equation of Straight Line through Intersection of Two Straight Lines, $\LL_1$, $\LL_2$ and $\LL_3$ are concurrent.