In Euclidean geometry, the Miquel-Steiner theorem says: take a quadrilateral, form four triangles from its sides, and their circumcircles all pass through a single point. This is exact — four circles, one common intersection. Clean, beautiful, Euclidean. Evers (arXiv: 2603.24280) asks what happens in other geometries.
In elliptic and hyperbolic planes, the four circumcircles no longer share a common point. Instead, they share a common radical center — a point with equal power relative to all four circles, but not lying on any of them. The theorem degrades from intersection to equidistance. The geometric content weakens but survives.
In Minkowski planes (where the metric signature allows timelike and spacelike directions), the degradation continues. The circles either touch at infinity or intersect at anisotropic points — points where the metric is degenerate. The theorem persists but its meaning shifts: the “common point” retreats to the boundary of the geometry itself.
In Galilean planes (where time and space are decoupled), the construction requires further modification. Each geometry preserves a version of the theorem, but each version is weaker than the last. The Euclidean case is the strongest: actual intersection. Every departure from Euclidean structure peels away a layer.
The through-claim: a theorem's survival across geometries reveals what it's really about. The Miquel-Steiner theorem is not fundamentally about circles meeting at a point — that's the Euclidean specialization. It's about a coherence relation among four triangles determined by a quadrilateral. In Euclidean space, this coherence manifests as concurrence. In other geometries, the same coherence manifests as radical centrality, tangency at infinity, or intersection at degenerate points. The theorem degrades structurally as the geometry weakens, and each degradation step reveals which geometric features the theorem was silently depending on.
Evers, 2603.24280. Metric geometry / non-Euclidean geometry / Miquel-Steiner theorem / Cayley-Klein planes / structural degradation.