Skip to content
DraconiticResearch
Search Sources API
197,768 studies31 sourcesUpdated 2 min ago
PreprintCC BY 4.0HAL

Biconic rotation of triangles

  • Evgeny A. Avksentyev
Open at sourcePDFCite

Abstract

If a circumscribed conic and an inscribed conic of a triangle are fixed, then, according to Poncelet's porism, there exists a “rotation” of this triangle during which it remains inscribed in the former and circumscribed about the latter. We call such a rotation biconic. The paper presents a number of geometric constructions with inscribed and circumscribed triangles of two conics. Using them, we provide a synthetic proof of Weill’s theorems on biconic rotation induced by a pencil of conics. Then, a more general dynamics of the biconical rotation of the main circumconic of the triangle — the circumscribed conic of the triangle passing through two fixed points — the contacts — is studied. All positions of the contacts are described for which the family ℳ𝒞 of main circumconics of poristic triangles forms a pencil of conics. It is then shown that for arbitrary contacts, the family ℳ𝒞 belongs to a certain bundle of conics. In particular, the circumcircles of all poristic triangles have a common radical center. Dual constructions are also presented.

Related

  1. Surgery Exact Triangles in Involutive Heegaard Floer Homology
    Kristen Hendricks, Jennifer Hom, Matthew Stoffregen +1
    Memoirs of the American Mathematical SocietyPeer-reviewedLicence not statedFull text31 citedGeometry & topologyOpenAlex
  2. Real monopole Floer homology and skein exact triangles
    Jiakai Li
    Quantum TopologyPeer-reviewedCC BY12 citedGeometry & topologyOpenAlex
  3. Napoleonic triangles on the sphere from similar chord triangles
    Quang Hung Tran
    arXivPreprintCC BY 4.0Full textGeometry & topologyarXiv
  4. Optimal discrepancy for triangles
    Gleb Smirnov
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsProbabilityarXiv
  5. Triangles and quadrilaterals suffice
    Mu-Tsun Tsai
    arXivPreprintarXiv licenceFull textGeometry & topologyarXiv

Cite

BibTeXRIS
BibTeX
@misc{avksentyev2026biconic,
  title = {{Biconic rotation of triangles}},
  author = {Evgeny A. Avksentyev},
  howpublished = {HAL},
  year = {2026},
  month = oct,
  url = {https://hal.science/hal-05838532v1},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Biconic rotation of triangles
AU  - Evgeny A. Avksentyev
PY  - 2026
DA  - 2026/10/25
PB  - HAL
UR  - https://hal.science/hal-05838532v1
AB  - If a circumscribed conic and an inscribed conic of a triangle are fixed, then, according to Poncelet's porism, there exists a “rotation” of this triangle during which it remains inscribed in the former and circumscribed about the latter. We call such a rotation biconic. The paper presents a number of geometric constructions with inscribed and circumscribed triangles of two conics. Using them, we provide a synthetic proof of Weill’s theorems on biconic rotation induced by a pencil of conics. Then, a more general dynamics of the biconical rotation of the main circumconic of the triangle — the circumscribed conic of the triangle passing through two fixed points — the contacts — is studied. All positions of the contacts are described for which the family ℳ𝒞 of main circumconics of poristic triangles forms a pencil of conics. It is then shown that for arbitrary contacts, the family ℳ𝒞 belongs to a certain bundle of conics. In particular, the circumcircles of all poristic triangles have a common radical center. Dual constructions are also presented.
ER  -