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W. Blaschke and H. R. Müller [4, p. 142] have given the following theorem as a generalization of the classic Holditch Theorem: Let $E/E^{\text{'}}$ be a 1-parameter closed planar Euclidean motion with the rotation number $\nu$ and the period $T$. Under the motion $E/E^{\text{'}}$, let two points $A=(0,0)$, $B=(a+b,0)\in E$ trace the curves $k_A,k_B\subset E^{\text{'}}$ and let $F_A,F_B$ be their orbit areas, respectively. If $F_X$ is the orbit area of the orbit curve $k$ of the point $X=(a,0)$ which is collinear with points $A$ and $B$ then $$F_X=\frac{[a{F}_B+b{F}_A]}{a+b}-\pi \nu ab.$$ In this paper, under the 1-parameter closed planar homothetic motion...

In this paper, under the one-parameter closed planar homothetic motion, a generalization of Holditch Theorem is obtained by using two different line segments (with fixed lengths) whose endpoints move along two different closed curves.

The aim of this paper is to investigate the orthogonality of vectors to each other and the Gram-Schmidt method in the Minkowski space ${ℝ}_2^3$. Hyperbolic cosine formulas are given for all triangle types in the Minkowski plane ${ℝ}_1^2$. Moreover, the Pedoe inequality is explained for each type of triangle with the help of hyperbolic cosine formulas. Thus, the Pedoe inequality allowed us to establish a connection between two similar triangles in the Minkowski plane. In the continuation of the study, the rotation...

We aim to introduce generalized quaternions with dual-generalized complex number coefficients for all real values $\alpha$, $\beta$ and $𝔭$. Furthermore, the algebraic structures, properties and matrix forms are expressed as generalized quaternions and dual-generalized complex numbers. Finally, based on their matrix representations, the multiplication of these quaternions is restated and numerical examples are given.