In this thesis, we will introduce the concept of hyperbolic geometry, which differs from Euclidean geometry in only one axiom. Hyperbolic geometry can be represented by several different models, such as the Beltrami-Klein model on the unit disk ${\mathbb D}$, the Poincaré model in the upper half-plane ${\mathbb H}$, and the Poincaré model on the unit disk ${\mathbb D}$, which are equivalent to each other. We will be most interested in the Poincaré model, as angles and circles in it correspond to those in Euclidean geometry. The seven circles theorem is a result of Euclidean geometry, which can be proven using classical tools. However, as we will see later, there is also a rather elegant proof of this theorem using hyperbolic geometry. Finally, this approach also allows for a generalization of the theorem.
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