Metallic Circles: Concyclic Tangency Points of Ford Circles over k-Fibonacci Quotients

@misc{pina2026metallic,
  title = {{Metallic Circles: Concyclic Tangency Points of Ford Circles over $k$-Fibonacci Quotients}},
  author = {Nicolás Piña},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.07141},
  eprint = {2611.07141},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.07141},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Metallic Circles: Concyclic Tangency Points of Ford Circles over k-Fibonacci Quotients
AU  - Nicolás Piña
PY  - 2026
DA  - 2026/11/04
PB  - arXiv
DO  - 10.48550/arxiv.2611.07141
UR  - https://arxiv.org/abs/2611.07141
AB  - Extending a result of Kocik, we show that the tangency points of the Ford circles built over quotients of k-Fibonacci numbers are concyclic: they lie on a circle centred on the horizontal axis, of radius √(k^2+4)/2, meeting it at the reciprocal of the metallic ratio σ_k and at −σ_k. Concyclicity turns out to be a quadratic identity of the sequence read in coordinates. In the Poincaré half-plane, this circle is a geodesic along which the tangency points advance with constant step 2lnσ_k.
ER  -