Closing Theorems for Circle Chains
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917933685407744 |
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| author | Hungerbühler, Norbert |
| author_facet | Hungerbühler, Norbert |
| contents | We consider closed chains of circles $C_1,C_2,\ldots,C_n,C_{n+1}=C_1$ such that two neighbouring circles $C_i,C_{i+1}$ intersect or touch each other with $A_i$ being a common point. We formulate conditions such that a polygon with vertices $X_i$ on $C_i$, and $A_i$ on the (extended) side $X_iX_{i+1}$, is closed for every position of the starting point $X_1$ on $C_1$. Similar results apply to open chains of circles. It turns out that the intersection of the sides $X_iX_{i+1}$ and $X_jX_{j+1}$ of the polygon lies on a circle $C_{ij}$ through $A_i$ and $A_j$ with the property that $C_{ij}, C_{jk}$ and $C_{ki}$ pass through a common point. The six circles theorem of Miquel and Steiner's quadrilateral Theorem appear as special cases of the general results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_15751 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Closing Theorems for Circle Chains Hungerbühler, Norbert General Mathematics 51M04, 51M15 We consider closed chains of circles $C_1,C_2,\ldots,C_n,C_{n+1}=C_1$ such that two neighbouring circles $C_i,C_{i+1}$ intersect or touch each other with $A_i$ being a common point. We formulate conditions such that a polygon with vertices $X_i$ on $C_i$, and $A_i$ on the (extended) side $X_iX_{i+1}$, is closed for every position of the starting point $X_1$ on $C_1$. Similar results apply to open chains of circles. It turns out that the intersection of the sides $X_iX_{i+1}$ and $X_jX_{j+1}$ of the polygon lies on a circle $C_{ij}$ through $A_i$ and $A_j$ with the property that $C_{ij}, C_{jk}$ and $C_{ki}$ pass through a common point. The six circles theorem of Miquel and Steiner's quadrilateral Theorem appear as special cases of the general results. |
| title | Closing Theorems for Circle Chains |
| topic | General Mathematics 51M04, 51M15 |
| url | https://arxiv.org/abs/2502.15751 |