Find the lengths of two trajectories that divide an arbitrary triangle into three equal angles

2021 ◽  
Vol 11 (4) ◽  
pp. 853-857
Author(s):  
Norkoziev Navruzbek Noralioglu
Keyword(s):  
Arbitrary Triangle

Stereographic Projection

2017 ◽  
Author(s):  
Glen Van Brummelen
Keyword(s):  
Stereographic Projection ◽  
Spherical Trigonometry ◽  
Arbitrary Triangle ◽  
Cesàro Method

This chapter deals with stereographic projection, which is superior to other projections of the sphere because of its angle-preserving and circle-preserving properties; the first property gave instrument makers a huge advantage and the second provides clear astronomical advantages. The earliest text on stereographic projection is Ptolemy's Planisphere, in which he explains how to use stereographic projection to solve problems involving rising times, suggesting that the astrolabe may have existed already. After providing an overview of the astrolabe, an instrument for solving astronomical problems, the chapter considers how stereographic projection is used in solving triangles. It then describes the Cesàro method, named after Giuseppe Cesàro, that uses stereographic projection to project an arbitrary triangle ABC onto a plane. It also examines B. M. Brown's complaint against Cesàro's approach to spherical trigonometry.

scholarly journals Faces of 2-Dimensional Simplex of Order and Chain Polytopes

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 851
Author(s):  
Aki Mori
Keyword(s):  
Partially Ordered Set ◽  
Ordered Set ◽  
Explicit Description ◽  
Dimensional Simplex ◽  
Partially Ordered ◽  
The Face ◽  
Finite Partially Ordered Set ◽  
Arbitrary Triangle

Each of the descriptions of vertices, edges, and facets of the order and chain polytope of a finite partially ordered set are well known. In this paper, we give an explicit description of faces of 2-dimensional simplex in terms of vertices. Namely, it will be proved that an arbitrary triangle in 1-skeleton of the order or chain polytope forms the face of 2-dimensional simplex of each polytope. These results mean a generalization in the case of 2-faces of the characterization known in the case of edges.

LOSSLESS SUBDIVISION-BASED MULTIRESOLUTION REPRESENTATION OF ARBITRARY TRIANGLE MESHES USING KITE TREES

2002 ◽  
Vol 13 (02) ◽  
pp. 243-260 ◽  
Author(s):  
WEI XU ◽  
RICHARD HAMMERSLEY ◽  
KAREN LU ◽  
DONALD FUSSELL
Keyword(s):  
Subdivision Surfaces ◽  
Tree Representation ◽  
Triangle Meshes ◽  
Surface Information ◽  
Mesh Topology ◽  
Multiresolution Representation ◽  
New Type ◽  
Arbitrary Triangle ◽  
Combine Surface ◽  
Regular Subdivision

We introduce a new type of subdivision-based multiresolution representation for triangle meshes, kite trees, which has the flexibility to represent arbitrary triangle meshes losslessly. We also develop an algorithm for extracting a balanced kite tree representation of an arbitrary input mesh that preserves mesh topology and regularities in the subdivision structure. Our scheme allows us to combine surface information automatically extracted from input data with algorithmically-generated information in a single multiresolution representation and to represent the results of adaptive refinement of regular subdivision surfaces.

On Napoleon triangles and propeller theorems

2003 ◽  
Vol 87 (508) ◽  
pp. 42-50
Author(s):  
Zvonko Čerin
Keyword(s):  
Equilateral Triangle ◽  
Triangle Geometry ◽  
Equilateral Triangles ◽  
Arbitrary Triangle

In this paper we shall consider two situations in triangle geometry when equilateral triangles appear and then show that they are closely related.In the first (known as the Napoleon theorem) equilateral triangles BCAT, CABT, and ABCT, are built on the sides of an arbitrary triangle ABC and their centroids are (almost always) vertices of an equilateral triangle ANBNCN (known as a Napoleon triangle of ABC; see Figure 1).

On n-sectors of the Angles of an Arbitrary Triangle

2020 ◽  
Vol 14 (4) ◽  
pp. 757-773
Author(s):  
Dongming Wang ◽  
Bo Huang ◽  
Xiaoyu Chen
Keyword(s):  
Arbitrary Triangle

Comparison Analysis of Fuzzy AHP and AHP in Multiple-Decision Making Problem Using Statistic Approach

2018 ◽  
Vol 1 (03) ◽  
Author(s):  
Linda Zhang
Keyword(s):  
Decision Making ◽  
Fuzzy Ahp ◽  
Simulated Data ◽  
Pairwise Comparison ◽  
Analytic Hierarchy ◽  
Weight Value ◽  
Value Range ◽  
The Difference ◽  
Arbitrary Triangle ◽  
Comparison Weight

Analytic Hierarchy Process (AHP) is a robust approach for decision making under complex criteria. Decision makers express their opinions differently and arbitrarily, giving rise to uncertainty in the ranking of alternatives. Fuzzy AHP was then developed and applied under those circumstances to reduce the uncertainty. This paper generates a string of randomly simulated data to represent completely arbitrary triangle fuzzy number, and based on these data compare Fuzzy AHP with classical AHP in statistical manner. Then the paper conducts a series of SPSS linear regressions for this comparison with two critical factors: the pairwise comparison weight value of AHP and the fuzzy value range of Fuzzy AHP. The regression shows how these two factors affect the differences between the two approaches. Results indicate that the pairwise comparison weight value of AHP significantly influences the difference while the fuzzy value range does not. In general, Fuzzy AHP narrows the final weights between each criterion, but in some extreme situations, this conclusion does not exist any more.

A Triangle Divided: Investigating Equal Areas

2000 ◽  
Vol 93 (7) ◽  
pp. 608-611
Author(s):  
Daniel Scher
Keyword(s):  
Simple Problem ◽  
Problem Statement ◽  
Education Development ◽  
Equal Area ◽  
Arbitrary Triangle ◽  
Curriculum Education ◽  
Development Center

Sometimes beneath a simple problem statement lies unexpected mathematical depth. Consider this challenge from the Connected Geometry curriculum (Education Development Center 2000): “Find as many ways as you can to divide an arbitrary triangle ABC into four equal-area triangles.” Before reading further, try solving the problem yourself.

Triangles meeting triangles

2014 ◽  
Vol 98 (543) ◽  
pp. 432-451
Author(s):  
Tony Crilly ◽  
Colin R. Fletcher
Keyword(s):  
List Type ◽  
Arbitrary Triangle

We consider two connected problems: •For a given but otherwise arbitrary triangle in the plane, to construct similar triangles which ‘meet’ this triangle.•To find the triangle so formed which has least area.1. Constructing a triangle which meets anotherThese problems beg the question of what is meant by ‘meet’ and we now aim to make this precise: Definition: A triangle XYZ will meet a given triangle ABC if on the triangle ABC, the vertex X lies on a line through AB, the vertex Y lies on a line through BC, and the vertex Z lies on a line through CA.When triangle XYZ is actually ‘in’ the triangle ABC, ‘meet’ is synonymous with the traditional ‘inscribe’ (such as in case (1) below). For ‘inscribe’ we understand that some of X, Y, Z may coincide with the vertices of ABC (such as case (2) below).More generally we use ‘meet’ to extend these possibilities by allowing XYZ to meet triangle ABC with its sides produced externally (such as case (3) below).

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