New Extensions of Napoleon’s Theorem to Higher Dimensions

نویسندگان

  • Mowaffaq Hajja
  • Horst Martini
  • Margarita Spirova
چکیده

If equilateral triangles are erected outwardly on the sides of any given triangle, then the circumcenters of the three erected triangles form an equilateral triangle. This statement, known as Napoleon’s theorem, and the configuration involved, usually called the Torricelli configuration of the initial triangle, were generalized to d-dimensional simplices (d ≥ 3) in [12]. It is obvious that for d ≥ 3 regular d-simplices cannot be erected on the facets of an arbitrary initial d-simplex S. Thus, instead of erecting such simplices, the authors of [12] used a related sphere configuration which also occurs in the planar situation. In the present paper, we give new d-dimensional analogues, mainly based on a higher dimensional Torricelli configuration constructed with the help of segments on lines through isogonal points and vertices of S. Interesting further properties of d-dimensional Torricelli configurations are obtained, too. MSC 2000: 51M04; 51M20; 51N20; 52B11

برای دانلود رایگان متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Some new extensions of Hardy`s inequality

In this study, by a non-negative homogeneous kernel k we prove some extensions of Hardy's inequalityin two and three dimensions

متن کامل

Some extensions of Darbo's theorem and solutions of integral equations of Hammerstein type

In this brief note,  using the technique of measures of noncompactness, we give some extensions of Darbo fixed point theorem. Also we prove  an existence result for a quadratic  integral equation of Hammerstein type on an unbounded interval in two variables  which includes several classes of nonlinear integral equations of Hammerstein type. Furthermore, an example is presented to show the effic...

متن کامل

Napoleon's Theorem and Generalizations Through Linear Maps

Recently J. Fukuta and Z. Čerin showed how regular hexagons can be associated to any triangle, thus extending Napoleon’s theorem. The aim of this paper is to prove that these results are closely related to linear maps. This reflects better the affine character of some constructions and gives also rise to a few new theorems. MSC 2000: 51M04

متن کامل

On the compactness property of extensions of first-order G"{o}del logic

We study three kinds of compactness in some variants of G"{o}del logic: compactness,entailment compactness, and approximate entailment compactness.For countable first-order underlying language we use the Henkinconstruction to prove the compactness property of extensions offirst-order g logic enriched by nullary connective or the Baaz'sprojection connective. In the case of uncountable first-orde...

متن کامل

Extensions of Some Fixed Point Theorems for Weak-Contraction Mappings in Partially Ordered Modular Metric Spaces

The purpose of this paper is to establish fixed point results for a single mapping in a partially ordered modular metric space, and to prove a common fixed point theorem for two self-maps satisfying some weak contractive inequalities.

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2008