pencil of lines


Aix+Biy+Ci=0 (1)

be equations of some lines.  Use the short notations  Aix+Biy+Ci:=Li.

If the lines  L1=0  and  L2=0  have an intersectionMathworldPlanetmath point P, then, by the parent entry (, the equation

k1L1+k2L2=0 (2)

with various real values of k1 and k2 can any line passing through the point P; this set of lines is called a pencil of lines.

Theorem.  A necessary and sufficient condition in to three lines


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


Proof.  If the line  L3=0  belongs to the fan of lines determined by the lines  L1=0  and  L2=0,  i.e. all the three lines have a common point, there must be the identity


i.e. there exist three real numbers k1, k2, k3, which are not all zeroes, such that the equation

k1L1+k2L2+k3L30 (3)

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


has nontrivial solutions k1,k2,k3. 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 k1,k2,k3.  Thus we can write the identic equation (3).  Let e.g.  k10.  Solving (3) for L1 yields


which shows that the line  L1=0  belongs to the fan determined by the lines  L2=0  and  L3=0; so the lines pass through a common point.


  • 1 Lauri Pimiä: Analyyttinen geometria.  Werner Söderström Osakeyhtiö, Porvoo and Helsinki (1958).
Title pencil of lines
Canonical name PencilOfLines
Date of creation 2013-03-22 18:09:03
Last modified on 2013-03-22 18:09:03
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Theorem
Classification msc 51N20
Related topic LineInThePlane
Related topic Determinant2
Related topic HomogeneousLinearProblem
Related topic Pencil2