În acest articol scoatem în evidență axa radicală a cercurilor Lemoine.
Laturile unui triunghi sunt împărțite de primul cerc Lemoine în segmente proporționale cu pătratele laturilor triunghiului.
Authors: Florentin Smarandache
Comments: 221 Pages.
The degree of difficulties of the problems is from easy and medium to hard. The solutions of the problems are at the end of each chapter. One can navigate back and forth from the text of the problem to its solution using bookmarks. The book is especially a didactical material for the mathematical students and instructors.
The first Lemoine circle divides the sides of a triangle in segments proportional to the squares of the triangle’s sides.
This paper presents rough netrosophic multiattribute decision making based on grey relational analysis. While the concept of neutrosophic sets is a powerful logic to deal with indeterminate and inconsistent data, the theory of rough neutrosophic sets is also a powerful mathematical tool to deal with incompleteness.
Authors: Joseph I. Thomas
Comments: 70 Pages.
This paper consolidates all the salient geometrical aspects of the principle of Polychronous Wavefront Computation. A novel set of simple and closed planar curves are constructed based on this principle, using MATLAB. The algebraic and geometric properties of these curves are then elucidated as theorems, propositions and conjectures.
Authors: Ben Steber
Comments: 4 Pages.
The author will demonstrate that the sums of odd numbers to an nth value equals that nth value squared. A geometric proof will be provided to demonstrate the principle of sum odds equaling squares.
Authors: Robert B. Easter
Comments: 62 Pages.
This paper introduces the Double Conformal / Darboux Cyclide Geometric Algebra (DCGA), based in the G8,2 Clifford geometric algebra. DCGA is an extension of CGA and has entities representing points and general Darboux cyclide surfaces in Euclidean 3D space. The general Darboux cyclide is a quartic surface. Darboux cyclides include circular tori and all quadrics, and also all surfaces formed by their inversions in spheres. Dupin cyclide surfaces can be formed as inversions in spheres of circular toroid, cylinder, and cone surfaces. Parabolic cyclides are cubic surfaces formed by inversion spheres centered on other surfaces. All DCGA entities can be conformally transformed in 3D space by rotors, dilators, translators, and motors, which are all types of versors. All entities can be inversed in general spheres and reflected in general planes. Entities representing the intersections of surfaces can be created by wedge products. All entities can be intersected with spheres, planes, lines, and circles. DCGA provides a higher-level algebra for working with 3D geometry in an object/entity-oriented system of mathematics above the level of the underlying implicit surface equations of algebraic geometry. DCGA could be used in the study of geometry in 3D, and also for some applications.