In this master’s thesis, we will discuss geometric constructions that can be drawn using a marked ruler. Such constructions were already used by ancient Greek mathematicians, who referred to them in Greek as neusis constructions. Unlike classical geometric constructions, which involve the use of a compass and an unmarked straightedge, in neusis constructions, we use a marked ruler with two marks. In this master's thesis, we will explore the methods of using a marked ruler and the types of constructions that can be performed with it. We will demonstrate that neusis constructions offer a wider range of possibilities compared to classical constructions. Our work will include basic geometric constructions, such as drawing parallels and perpendiculars to given lines, as well as constructions providing solutions to three famous ancient Greek geometric problems, namely angle trisection, doubling the cube, and construction of a regular heptagon. We will examine how these constructions are justified by fundamental geometric theorems, and we will support the explanations with algebraic arguments as well. We will prove that equations of the fourth degree can be solved using neusis constructions. We will pay special attention to the drawing of constructions that are impossible to construct using classical geometric tools.
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