On a Configuration with Circle-Conic Tangency and Sharygin Points

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.