The six circles theorem of Miquel and Steiner's quadrilateral Theorem appear as special cases of the general results.
📌 3 Key Takeaways
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Objective & Hypothesis: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.
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Key Findings & Data: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 $C_1$.
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Conclusion & Impact:The six circles theorem of Miquel and Steiner's quadrilateral Theorem appear as special cases of the general results.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Closing theorems · circle chains · Miquel’s six circles theorem
Abstract
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.
Hungerbühler, N. (2025). Closing Theorems for Circle Chains. International Electronic Journal of Geometry, Volume: 18 Issue: 2. https://doi.org/10.36890/iejg.1624214
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