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# pencil of lines

Let

$\displaystyle A_{i}x+B_{i}y+C_{i}=0$ | (1) |

be equations of some lines. Use the short notations $A_{i}x+B_{i}y+C_{i}\,:=\,L_{i}$.

If the lines $L_{1}=0$ and $L_{2}=0$ have an intersection point $P$, then, by the parent entry, the equation

$\displaystyle k_{1}L_{1}+k_{2}L_{2}=0$ | (2) |

with various real values of $k_{1}$ and $k_{2}$ can represent any line passing through the point $P$; this set of lines is called a pencil of lines.

Theorem. A necessary and sufficient condition in order to three lines

$L_{1}=0,\quad L_{2}=0,\quad L_{3}=0$ |

pass through a same point, is that the determinant formed by the coefficients of their equations (1) vanishes:

$\left|\begin{matrix}A_{1}&B_{1}&C_{1}\\ A_{2}&B_{2}&C_{2}\\ A_{3}&B_{3}&C_{3}\end{matrix}\right|=\left|\begin{matrix}A_{1}&A_{2}&A_{3}\\ B_{1}&B_{2}&B_{3}\\ C_{1}&C_{2}&C_{3}\end{matrix}\right|=0.$ |

Proof. If the line $L_{3}=0$ belongs to the fan of lines determined by the lines $L_{1}=0$ and $L_{2}=0$, i.e. all the three lines have a common point, there must be the identity

$L_{3}\equiv L_{1}+L_{2},$ |

i.e. there exist three real numbers $k_{1}$, $k_{2}$, $k_{3}$, which are not all zeroes, such that the equation

$\displaystyle k_{1}L_{1}+k_{2}L_{2}+k_{3}L_{3}\equiv 0$ | (3) |

is satisfied identically by all real values of $x$ and $y$. This means that the group of homogeneous linear equations

$\displaystyle\begin{cases}k_{1}A_{1}+k_{2}A_{2}+k_{3}A_{3}=0,\\ k_{1}B_{1}+k_{2}B_{2}+k_{3}B_{3}=0,\\ k_{1}C_{1}+k_{2}C_{2}+k_{3}C_{3}=0\end{cases}$ |

has nontrivial solutions $k_{1},\,k_{2},\,k_{3}$. By linear algebra, it follows that the determinant of this group of equations has to vanish.

Suppose conversely that the determinant vanishes. This implies that the above group of equations has a nontrivial solution $k_{1},\,k_{2},\,k_{3}$. Thus we can write the identic equation (3). Let e.g. $k_{1}\neq 0$. Solving (3) for $L_{1}$ yields

$L_{1}\equiv-\frac{k_{2}L_{2}+k_{3}L_{3}}{k_{1}},$ |

which shows that the line $L_{1}=0$ belongs to the fan determined by the lines $L_{2}=0$ and $L_{3}=0$; so the lines pass through a common point.

# References

- 1 Lauri Pimiä: Analyyttinen geometria. Werner Söderström Osakeyhtiö, Porvoo and Helsinki (1958).

## Mathematics Subject Classification

51N20*no label found*

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