Holditch-type theorems for the polar moment of inertia of the closed orbit curve under planar homothetic motions

H. Holditch [1] have given the following important theorem in 1858:If the endpoints of a line segment with fixed length are rotated once along an oval (Eilinie) in the Euclidean plane, then a given fixed point on line segment describes a closed, not necessarilly convex, curve. The area of the Holditch-Ring bounded by the oval and curve isW. Blaschke and H. R. Müller [3] have given the following generalization of the classical Holditch Theorem:Under 1-parameter closed planar motions with the rotation number, if the endpoints and on a line segment trace the closed orbit curves and, respectively, the point which is collinear with points and traces a closed, not necessarilly convex, curve. Let and be the orbit areas of the orbit curves and, respectively. Then, there is the equationIn this study, under the homothetic motions, we investigated the obtained results which Holditch-Type Theorems for the polar moments of inertia of three noncollinear points given by Yüce, S., Düldül, M. ve Kuruoğlu, N. [7].