This diploma thesis seeks to discuss and prove basic tasks and basic constructions of plane geometry by detailed knowledge of linear algebra. First of all, it is demonstrated what those constructions actually consist of and why certain relations are valid. Therefore, Euclids and Hilberts axiomatic systems are introduced. Euclid was the first mathematician to apply the deductive method. Additionally, he also collected the universe knowledge of geometry in his compilation The Elements. David Hilbert finally defined plane geometry via a system of axioms, which turned out to be complete and consistent, such as proven in his book The Foundations of Geometry.As the plane can also be understood as a vector space, the most important components of linear algebra are introduced. These components are necessary to proof the mentioned constructions in an analytic way throughout this thesis.Furthermore, basic constructions and their applications are presented by trying not to use tools except for compass and spacer. Even cases of inaccessible points are considered. The constructions, as well as the theorems, which are elementary for their realization, are formulated and established.Finally, the Inscribed Angle Theorem and its reversal are examined with algebraic methods. The Thales Theorem as a special case of the Inscribed Angle Theorem is also essential for some of the previously discussed tasks.