A classical porism about two nonintersecting circles (an annulus) stating that if there exists a closed chain of n pairwise tangent circles inscribed between the two given circles for one choice of a starting circle, then such a closed chain exists for every choice of starting circle in the annular region.

Principle

An incidence invariance: the closure property of a tangent circle chain between two fixed coaxal circles is independent of the initial tangent circle once one closed chain exists, reflecting an underlying integrable geometric constraint.

Demonstration

Construct two nonintersecting concentric circles and inscribe a chain of equal tangent circles around the inner circle touching the outer circle; when a closed n‑chain exists for one position it can be rotated continuously to produce closed chains for all starting positions in the annulus.

Misapplication

Assuming the porism holds when the outer and inner circles intersect, are tangent, or when the inserted circles are not all tangent consecutively; or treating the porism as a statement about arbitrary nested curves rather than coaxal circles.

Consequence

Yields a one-parameter family of closed tangent chains and shows that certain circle‑packing configurations are rigid in closure behavior; it informs constructions in inversion geometry and links to Möbius transformations.

Reversal

The inverse viewpoint emphasizes failure of closure: if no closed chain exists for any starting circle, then no chain closes; reversing the porism highlights dependence on relative circle positions rather than arbitrary starting data.

Boundary

Applies to configurations of two disjoint circles in the Euclidean plane (or their images under Möbius transformations); exclusions include intersecting or coincident circles, noncircular boundaries, or chains that do not preserve tangency order.

Semantic Tension

Nearby concepts include Apollonian circle packings and Poncelet porisms; the tension lies between Steiner's closure invariance (chains of tangent circles) and Poncelet's polygonal closure (polygons tangent and circumscribed) which are analogous but differ in objects and invariants.

Synthesis

Steiner Porism encapsulates a geometric rigidity: when one closed tangent circle chain between two fixed nonintersecting circles exists, the annulus admits a continuous family of such closed chains, revealing an invariant closure property under moving the starting circle.