A pencil is a set of geometric objects, usually either congruent or similar to each other, that share a common incidence property. Below are some of the most commonly encountered pencils:

1. A pencil of lines usually means a set of straight lines that are incident with one point. If the lines all lie in the same plane, the pencil is sometimes called a flat pencil.
2. In some cases, a pencil of lines denotes a set of parallel lines in a plane. If a point of infinity is added to the plane, then we are back to the previous example.
3. A pencil of circles can mean that either these circles all intersect at exactly one point (or share the same tangent line)
4. A pencil of circles can also mean that the circles have two common points of intersection.

Bibliography

1

E. Artin, Geometric Algebra, Wiley-Interscience, Reprint (1988).

2

H. S. M. Coxeter, Projective Geometry, Springer-Verlag, 2nd Edition (2003).