On a Configuration with Circle-Conic Tangency and Sharygin Points
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908597410070528 |
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| author | Kim, Petr Makoian, Georgii |
| author_facet | Kim, Petr Makoian, Georgii |
| contents | This work studies circle-geometry methods through their application to a main theorem about circles tangent twice to a conic. The authors investigate the Sharygin point -- a point lying in the pencil of two non-intersecting circles -- and explore its properties. These properties are applied to solve several olympiad problems, such as problems from MGO 2024 and the Croatian IMO selection. The paper also presents a simplified version of the main theorem and gives two different proofs: one using Sharygin points and another using Lobachevsky (hyperbolic) geometry. The article explores relation between Lorenz transformations of Minkowski space-time and a certain transformation of circles in hyperbolic geometry. The paper demonstrates the effectiveness of combining classical planimetry with ideas of non-Euclidean geometry for solving difficult problems involving circle tangencies. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_14508 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a Configuration with Circle-Conic Tangency and Sharygin Points Kim, Petr Makoian, Georgii Metric Geometry 51M04, 51A20 (Primary), 51M10, 51N20, 51A05 (Secondary) This work studies circle-geometry methods through their application to a main theorem about circles tangent twice to a conic. The authors investigate the Sharygin point -- a point lying in the pencil of two non-intersecting circles -- and explore its properties. These properties are applied to solve several olympiad problems, such as problems from MGO 2024 and the Croatian IMO selection. The paper also presents a simplified version of the main theorem and gives two different proofs: one using Sharygin points and another using Lobachevsky (hyperbolic) geometry. The article explores relation between Lorenz transformations of Minkowski space-time and a certain transformation of circles in hyperbolic geometry. The paper demonstrates the effectiveness of combining classical planimetry with ideas of non-Euclidean geometry for solving difficult problems involving circle tangencies. |
| title | On a Configuration with Circle-Conic Tangency and Sharygin Points |
| topic | Metric Geometry 51M04, 51A20 (Primary), 51M10, 51N20, 51A05 (Secondary) |
| url | https://arxiv.org/abs/2510.14508 |