Poncelet–Steiner theorem

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 states that whatever can be constructed by straightedge and compass together can be constructed by straightedge alone, provided that a single circle and its centre are given. This theorem is related to the rusty compass equivalence.

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. All constructions remain possible, though it is naturally understood that circles and their arcs cannot be drawn without the compass. This means only that the compass may be used for aesthetic purposes, rather than for the purposes of construction. 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.

Constructions carried out in adherence with this theorem - relying solely on the use of a straightedge tool without the aid of a compass - are known as Steiner constructions. Steiner constructions may involve any number of circles, including none, already drawn in the plane, with or without their centers. They may involve all manner of unique shapes and curves preexisting in the plane, also, provided that the straightedge tool is the only physical tool at the geometers disposal. Whereas the Poncelet-Steiner theorem stipulates the existence of a circle and its center, and affirms that a single circle is equivalent to a compass.