Editing Poncelet–Steiner theorem (section)

== Related constructs ==

Related constructs that link to this article.

=== Steiner constructions ===
The term '''Steiner construction''' typically refers to any geometric construction that utilizes the straightedge tool only, and is sometimes simply called a ''straightedge-only construction'', named for Jakob Steiner who studied the subject. As a restricted construction [[paradigm]], no stipulations are made about what geometric objects already exist in the plane or their relative placement; any such conditions are postulated ahead of time. Also, no implications are made about what is or is not possible to construct. Constructions carried out in adherence with the Poncelet-Steiner theorem - relying solely on the use of a straightedge tool without the aid of a compass - are therefore a particular subset of ''Steiner constructions''.

Whereas Steiner constructions study the straightedge tool, the Poncelet-Steiner theorem stipulates the existence of a circle with its center, and affirms that a single circle is equivalent to a compass. Broadly, Steiner constructions may involve any number of circles, including none, already drawn in the plane, with or without their centers. They may also involve all manner of unique shapes and curves preexisting in the plane, provided that the straightedge tool is the only physical tool at the geometer's disposal.

Therefore, all constructions adhering to the Poncelet-Steiner theorem are Steiner constructions, though not all Steiner constructions abide by the strict condition of there being only one circle with its center provided in the plane. The Poncelet-Steiner theorem does not require an actual compass - it is presumed that the circle preexists in the plane - therefore all constructions herein demonstrating the Poncelet-Steiner theorem are Steiner constructions. The single arbitrary circle (with its center), which is postulated in the Poncelet-Steiner theorem, is therefore the minimal amount of information required to allow Steiner constructions to recover the constructive power and versatility of the traditional compass-straightedge paradigm.

=== Rusty compass ===
The '''rusty compass''' describes a compass whose hinge is so rusted as to be fused such that its legs - the needle and pencil - are unable to adjust width. In essence, it is a compass whose distance is fixed, and which draws circles of a predetermined and constant, but arbitrary radius. Circles may be drawn centered at any arbitrary point, but the radius is unchangeable.

As a restricted construction paradigm, the ''rusty compass constructions'' allow the use of a straightedge and the fixed-width compass. The rusty compass equivalence:

:''All points necessary to uniquely describe any compass-straightedge construction may be achieved with a straightedge and fixed-width compass.''

It is naturally understood that the arbitrary-radius compass may be used for aesthetic purposes; only the arc of one specific predetermined fixed-width compass may be used for construction.

==== As the rusty compass relates to the theorem ====
In some sense, the rusty compass is a generalization and simplification of the Poncelet-Steiner theorem. Though not more powerful, it is certainly more convenient. The Poncelet-Steiner theorem requires a single circle with arbitrary radius and center point to be placed in the plane. As it is the only drawn circle, whether or not it was drawn by a rusty compass is immaterial and equivalent. The benefit of general rusty compass constructions, however, is that the compass may be used repeatedly to redraw circles centered at any desired point, albeit with the same radius, thus simplifying many constructions. Naturally if all constructions are possible with a single circle arbitrarily placed in the plane, then the same can surely be said about a straightedge and rusty compass, with which at least one circle may be arbitrarily placed. 

It is known that a straightedge and a rusty compass is sufficient to construct all that is possible with straightedge and standard compass - with the implied understanding that circular arcs of arbitrary radii cannot be drawn, and only need be drawn for aesthetic purposes rather than constructive ones.  Historically this was proven when the Poncelet-Steiner theorem was proven, which is a stronger result. The rusty compass, therefore, is no weaker than the Poncelet-Steiner theorem. It should also be intuitively clear that a rusty compass cannot be any stronger than a standard compass. As the rusty compass is at least as strong as the circle and no stronger than the standard compass, and the latter two are proven equivalent by the Poncelet-Steiner theorem, so too must the rusty compass be equivalent.

The Poncelet-Steiner theorem reduces Ferrari's rusty compass equivalence, a claim at the time, to a single-use compass:
:''All points necessary to uniquely describe any compass-straightedge construction may be achieved with only a straightedge, once the first circle has been placed.''
The Poncelet-Steiner theorem takes the rusty compass scenario, and breaks the compass completely after its first use.

There is a subtle distinction, however, which makes the single use rusty compass slightly more useful. With a single-use compass, the [[geometer]] may place the first circle arbitrarily, as it suits convenience, even after other constructions have begun. The Poncelet-Steiner theorem, on the other hand, presumes the geometer has no control over the placement of the circle or over the constructions undertaken, making the two independent.