N. Kuruoglu, Ondokuz Mayis University, Science and Arts Faculty, Department of Mathematics, Kurupelit 55139, Samsun Turkey, e-mail: kuruoglu@omu.edu.tr; S. Yuce, Ondokuz Mayis University, Science and Arts Faculty, Department of Mathematics, Kurupelit 55139, Samsun Turkey, e-mail: salimy@omu.edu.tr
Abstract: W. Blaschke and H. R. Muller [4, p. 142] have given the following theorem as a generalization of the classic Holditch Theorem: Let $E/E'$ be a 1-parameter closed planar Euclidean motion with the rotation number $\nu$ and the period $T$. Under the motion $E/E'$, let two points $A = (0, 0)$, $B = (a + b, 0) \in E$ trace the curves $k_A, k_B \subset E'$ 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 = {[aF_B + bF_A] \over a + b} - \pi\nu a b.
In this paper, under the 1-parameter closed planar homothetic motion with the homothetic scale $ h = h (t)$, the generalization given above by W. Blaschke and H. R. Muller is expressed and
F_X = {[aF_B + bF_A]\over a + b} - h^2 (t_0) \pi\nu a b,
is obtained, where $\exists t_0 \in[0, T]$.
Keywords: Holditch Theorem, homothetic motion, Steiner formula
Classification (MSC 2000): 53A17
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