Editing Poncelet–Steiner theorem (section)

{{short description|Universality of construction using just a straightedge and a single circle with center}}
[[File:Steiner construction of a parallel to a diameter.gif|thumb|''To draw the parallel (h) to a diameter g through any given point P.'' Chose auxiliary point C anywhere on the straight line through B and P outside of BP. (Steiner)]]
In the branch of [[mathematics]] known as [[Euclidean geometry]], the '''Poncelet–Steiner theorem''' is one of several results concerning [[compass and straightedge]] constructions having additional restrictions imposed on the traditional rules. This result, related to the '''rusty compass''' equivalence and to '''Steiner constructions''', states that whatever can be constructed by [[straightedge]] and [[compass (drafting)|compass]] together can be constructed by straightedge alone, provided that a single [[circle]] and its [[centre_(geometry)|centre]] are given:

: ''Any Euclidean construction, insofar as the given and required elements are points (or lines), if it can be completed with both the compass and the straightedge together, may be completed with the straightedge alone provided that no fewer than one circle with its center exist in the plane.''

Though a compass can make constructions significantly easier, it is implied that there is no functional purpose of the compass once the first circle has been drawn; the compass becomes redundant. All constructions remain possible, though it is naturally understood that circles and their [[circular arc|arcs]] cannot be drawn without the compass. All points that uniquely define a construction, which can be determined with the use of the compass, are equally determinable without, albeit with greater difficulty.

This means only that the compass may be used for [[aesthetic]] purposes, rather than for the purposes of construction. In other words, the compass may be used ''after'' all of the key points are determined, in order to "fill-in" the arcs purely for visual or artistic purposes, if it is desirable, and not as a necessary step toward construction. Nothing essential for the purposes of geometric construction is lost by neglecting the construction of circular arcs.