The main topic of this master thesis are geometric applications of the numerical range of the operator, which specifically deals with the 2 × 2 complex matrices. The first chapter defines a numerical range and gives some of its properties that are crucial in geometric proofs. Then, in the next chapter, the convexity property of the numerical range of the matrix is described in detail, supported by the Elliptical Range Theorem. It is also shown that the Elliptical Range Theorem enables a common approach to the proofs of the following theorems and each of them in its own way includes the notion of an ellipse. In the third chapter several versions of Siebeck’s Theorem are shown, that describe the connection between a complex polynomial of degree 3 and its derivative. Furthermore, the fourth chapter presents the connection between Blaschke’s product from complex analysis and Poncelet’s Theorem from the field of projective geometry. Also, here is the focus on the Blaschke’s 3-ellipse and the Poncelet’s 3-ellipse which are essentially equal. Finally, connection between these theorems that seem unrelated at first glance will be given.