Minimal Sum of Powered Distances from the Sides of a Triangle
In this paper we apply the KKT conditions to find the minimal sum of powered distances from the sides of an arbitrary triangle.
math.OC↗
arXiv subjects
Elias Abboud
Publications and source records attributed to Elias Abboud.
On Word Equations Originated from Discrete Dynamical Systems Related to Antisymmetric Cubic Maps With Some Applications
In this article, we solve some word equations originated from discrete dynamical systems related to antisymmetric cubic map. These equations emerge when we work with primitive and greatest words. We conclude with some applications
math.DS↗
Loci of Points Inspired by Viviani's Theorem
We consider loci of points such that their sum of distances or sum of squared distances to each of the sides of a given triangle is constant. These loci are inspired by Viviani's theorem and its extension. The former locus is a line segment or the whole triangle and the latter locus is an ellipse.
math.HO↗
Viviani's Theorem and its Extensions Revisited; Canghareeb, Sanghareeb and a New Definition of the Ellipse
Canghareeb and Sanghareeb are described to be "geometric creatures" living in the plane under certain conditions relative to a triangle. They have common and different features. The former lives on a line segment inside a triangle and the latter lives along an ellipse. Their story ends with a new definition of the ellipse.
math.MG↗
On Routh-Steiner Theorem and Generalizations
Following Coxeter we use barycentric coordinates in affine geometry to prove theorems on ratios of areas. In particular, we prove a version of Routh-Steiner theorem for parallelograms.
math.MG↗
On Viviani's Theorem and its Extensions
Viviani's theorem states that the sum of distances from any point inside an equilateral triangle to its sides is constant. We consider extensions of the theorem and show that any convex polygon can be divided into parallel segments such that the sum of the distances of the points to the sides on each segment is constant. A polygon possesses the CVS property if the sum of the distances from any inner point to its sides is constant. An amazing result, concerning the converse of Viviani's theorem is deduced; Three non-collinear points which have equal sum of distances to the sides inside a convex polygon, is sufficient for possessing the CVS property. For concave polygons the situation is quite different, while for polyhedra analogous results are deduced.
math.MG↗
Algorithms for Producing and Ordering Lexical and Nonlexical Sequences out of one Element
This paper deals with algorithms for producing and ordering lexical and nonlexical sequences of a given degree. The notion of "elementary operations" on positive integral sequences is introduced. Our main theorem answers the question of when two lexical sequences are adjacent.
math.CO↗