Editing Poncelet–Steiner theorem (section)

=== Pencils of lines or of circles ===
{{main|Pencil (geometry)}}
The term '''pencil''' refers to a [[set_(mathematics)|set]] of geometric objects that all share in a common property which is uniquely identifiable by precisely two of its elements, and is a term usually only employed in geometric contexts. The property in question is usually expressed as a geometric object also. For example, in the case of a ''pencil of lines'', which is a set containing only lines, the property is typically that of passing through the same point, also called concurrence. Two lines will intersect at a point - even if the point is "at infinity" in the case of parallel lines - where the point of concurrence (i.e. intersection) is the geometric object defining the property; any other line that also intersects at the same point is therefore in the pencil, and conversely those lines that do not are not in the pencil.

In the case of a ''pencil of circles'', a common [[Radical_axis#Coaxal_circles|coaxial system]] - that is, having the same radical axis (which is a line) - is the usual interpretation. Any other circle is included in the pencil if it shares the same radical axis with any other pair of circles already in the pencil. Indeed, a ''pencil of points'' typically refers to the set of all points on a given line, as any and every two points from this set will define the same line.

Though these are the usual meanings, any property the geometer chooses is valid, provided that it takes two elements - no more or less - to establish the underlying set. In essence, a pencil is an entire set of (potentially infinite) geometric objects which are wholly defined by any two, and every two, distinct members from its set. Any two like-objects therefore define the entire set to which other like-objects do or do not belong.

==== Historical publications ====

Pencils are a common theme in many geometry publications throughout history, though the term is less commonly used today. This article does not explicitly refer to pencils, though some of the constructions found herein, and in projective geometry more broadly, do in fact implicitly use the notion of a pencil, often by different terminology or by explicitly referring to the underlying property. The term is herein defined for the readers convenience due to its common usage within many of the cited references that are used to support the content of this article, wherein the term is rarely defined.

==== Generalizations of the pencil ====
Pencils have been generalized to higher dimensions as well (e.g. a ''pencil of planes''). For contrast, the term ''flat pencil'' refers to pencils in a two-dimensional space, and ''spatial pencil'' for three-dimensions.

The notion of pencils can also be generalized such that more than two elements establish the set. A set of geometric objects defined by precisely three of its elements, no more or less, is called a ''bundle''. One example of a ''bundle of points'' is the set of points on the arc of a circle.  Similarly, though there is no specialized term for it, five points in the plane define a [[conic section]], any five points on which may be used to establish the same curve.