The Steiner formula and the Holditch-type theorems in the generalized complex plane

This thesis consists of six chapters. In the first chapter, primarily, the classical Holditch theorem in Euclidean plane is expressed. Afterwards, summary of literature together with a general evaluation of some significant studies reletad to the Holditch theorem and the Holditch-type theorems is given. The second chapter consists of four subsections. Firstly, some basic notions of dimensional Euclidean space are given. Then, the one-parameter planar motions and the Holditch-type theorems in complex, Galilean and hyperbolic planes are introduced, respectively. The third chapter consists of two subsections. In the first subsection, the generalized complex number system is represented. Moreover, the geometric exposition and algebraic properties of this generalized complex number system are expressed. In the second subsection, the one-parameter planar motions in the generalized complex plane are given. The fourth and fifth chapters are the original part of this thesis. The fourth chapter consists of two subsections. First of all, by calculating the area of orbit curve in the fixed plane drawn by the fixed point in the moving generalized complex plane, the Holditch theorem that gives the relationship between the orbit curves drawn by linear three points is given. Then, by considering non-linear three points in generalized complex plane, a generalization of the Holditch theorem, which was given in the first subsection is obtained. In the fifth chapter, the polar moment of inertia of the orbit curve drawn by the fixed point in the generalized complex plane is calculated. Finally, the Holditch-type theorems for linear and non-linear three points are proved. In the sixth chapter, an evaluation of this thesis has been made discussed. Keywords: The generalized complex plane, the one-parameter planar motions, Steiner formula, Holditch-type theorems