Властивості бідотичного кола

This paper explores the properties of the bitangent circle, which is defined as a circle that is tangent to two sides of a triangle and to the circumcircle of the triangle. The problem related to such a circle was first proposed at the 20th International Mathematical Olympiad (1976, Bucharest) for an isosceles triangle. However, it was later discovered that the problem's condition is general for any triangle. The study of this problem attracted the attention of mathematicians from various countries, and numerous attempts at a proof emerged. This paper presents a new approach to solving the problem using formulas that simplify the proof compared to other methods, such as Archimedes' lemma. Additionally, the paper examines specific properties of the bitangent circle, which can form the basis for problems of increased difficulty. The content of the paper will interest mathematicians, educators, students, and all those passionate about geometry.