Poncelet's closure theorem

Poncelet's closure theorem states that if one polygon with a fixed number of sides is inscribed in one conic and circumscribed about another, then every starting point on the first conic produces a closed polygon of that kind.

Poncelet map: A map on a conic that sends each point to the next vertex selected by a tangent to a second conic. Its iterates encode the successive vertices whose eventual return the theorem guarantees.
Conic section: A plane curve obtained by intersecting a cone with a plane, including ellipses, parabolas, and hyperbolas. The theorem concerns pairs of conics and the polygons formed between them.
Jean-Victor Poncelet: A French mathematician and engineer whose work helped establish projective geometry. His 1822 treatise developed the geometric setting associated with the closure theorem.
Poncelet triangle: A triangle inscribed in one conic and tangent to a second conic. The three-sided case is the simplest nondegenerate polygon covered by the theorem.
Cayley–Bacharach theorem: A theorem about how two plane curves passing through all but one point of a complete intersection constrain the remaining point. Algebraic-geometric arguments related to closure connect with incidence principles of this kind.
Projective geometry: The study of properties preserved by projective transformations, including incidence and cross-ratio. Projective transformations simplify the conics while preserving the incidence structure behind closure.
Inscribed polygon: A polygon whose vertices lie on a specified curve. The polygon’s vertices must lie on the outer conic.
Porism: A mathematical proposition stating that if a condition is possible under certain constraints, it holds for infinitely many cases. Poncelet’s closure result is a celebrated example of a porism.
Poncelet pentagon: A five-sided polygon inscribed in one conic and tangent to another. It illustrates that closure applies beyond the triangular case.
Abel's theorem: A theorem characterizing when a divisor on an algebraic curve is the divisor of a meromorphic function. The elliptic-curve formulation translates polygon closure into a condition on divisor classes.

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Steiner conicRelated: It is a related projective conic phenomenon built from correspondence and incidence.