Sides Of Triangle Research Articles

The Various Radii Circle Packing Problem in a Triangle

Malfatti’s problem is the problem of fitting three circles into a triangle such that they are tangent to each other and each circle is also tangent to a pair of the triangle’s sides. This problem has been extended to include Tn = 1 + 2 + … + n circles inside the triangle with special tangency properties among the circles and triangle sides; this problem is referred to as the extended Malfatti problem or the Tri(Tn) problem. In the extended Malfatti problem, the number of circles in the triangle is a triangle number because the tangency properties between the internal circles and the three sides of the triangle have a special type of structure; that is, the corner circle is tangent to two sides of the triangle and two other circles, the boundary circles are tangent to one side of the triangle and four other circles, and the inner circles are always tangent to six other circles. The circles we find in the extended Malfatti problem have the following property: the smallest and largest radii of the circles differ to a great extent. In the study presented herein, we propose algorithms to solve the problem that the tangency properties between the circles and the sides of the triangle are not fixed, so that the number of circles in the triangle is not necessarily a triangle number. The purpose of this change is to attempt to establish the radii of the circles in the triangle within a small range.

Equilibrium and self-assembly of Janus particles at liquid-liquid interfaces for the film formation

This article investigates the equilibrium arrangement, self-assembly process, and subsequent curing of amphiphilic snowman-shaped Janus particles at the oil-water interface. The independent Janus particles are in vertical equilibrium state and the contact position of the oil-water interface is at the largest cross section of the particle’s hydrophobic phase. Under the effect of the surface tension and the adsorption of materials, Janus particles may form particle combinations including the particle pairs and the particle triangle, whose inner and outer sides have the liquid surface exhibiting completely opposite contact angles. Particle combinations form stable parallel double-chain structures with diverse shapes after the self-assembly process. However, the single Janus particles attain a state of mechanical equilibrium under the influence of surrounding particles, enabling them to assemble into regular array structures. The relationship of interfacial tension coefficient between phases can be changed by adjusting the oil-water system, which leads to variations in the self-assembly speed and the final arrangement results. The thin-film with uniformly distributed vertical particles is achieved by replacing the underlying deionized water with a curing agent. Based on the understanding of the interactions between irregularly shaped Janus particles at the oil-water interface, it will be convenient to achieve the controllable self-assembly and widely applications of these particles.

Side-side-angle with fixed angles and the cyclic quadrilateral

Among the triangle congruence axioms, the side-side-angle (SsA) axiom states that two triangles are congruent if and only if two pairs of corresponding sides and the angles opposite the longer sides are equal. The modification of the SsA axiom provides a construction with two triangle sequences . We require that the opposite angles of the equivalent shorter sides be fixed and the longer sides be equal. The locus of the intersection points of other sides of triangles is derived to be a hyperbola, and in a generalized form defined by a complete quadrilateral, it is a conic section.

Triangle side ratio method for particle angularity characterization: from quantitative assessment to classification applications

Triangle side ratio method for particle angularity characterization: from quantitative assessment to classification applications

Micro Learning-based Module: A Teaching Tool to Enhance Students’ Competence in Grade 9 Trigonometry

Globally, as the education field gear towards recovery from the Covid 19 pandemic, there is a must to adapt teaching strategies that will equip the teachers to continuously provide quality and engaging activities in a simplified manner that will bring out the best of the students. This paper describes the effectiveness of microlearning-based module as a teaching tool in enhancing students’ competence in Trigonometry. Data analysis gave results that there was mean difference of of 4.46 in illustrating, 5.02 for identifying, and 4.5 in solving after being exposed to the created learning material. The posttest result suggests that microlearning-based useful tool for enhancing students’ competence of the 9th – grade students in Trigonometry. Level. The hypotheses that claimed there was no significant difference between mean scores in the pretest and post- test of the three competencies in Trigonometry was negated. According to the results, students’ competence in Trigonometry were enhanced in terms of mean scores. Future student performance in the three competencies in Trigonometry can be projected to be better and more satisfying if this trend holds if it does. The created learning material can be credited with enhancing students' competence in illustrating the following topics: the six trigonometric ratios, the trigonometric ratios of special angles, and angle of elevation and angle of depression. Also, the microlearning module can improved the competency of identifying the trigonometric ratio that can be used in relating the angles and sides of a right triangle. The designed learning material can help the students enrich their solving skills such as solving the missing parts of a right triangle, evaluating trigonometric expressions of involving special angles, and problem solving involving right triangle, angle of elevation, and angle of depression. Hence, it is highly recommended that the school use this learning material to encourage effective teaching and quality learning.

On Some Properties of the Equilateral Triangles with Vertices Located on the Support Sides of a Triangle

The possible positions of an equilateral triangle whose vertices are located on the support sides of a generic triangle are studied. Using complex coordinates, we show that there are infinitely many such configurations, then we prove that the centroids of these equilateral triangles are collinear, defining two lines perpendicular to the Euler’s line of the original triangle. Finally, we obtain the complex coordinates of the intersection points and study some particular cases.

Surface stress redistribution and thermal performance of tempered vacuum glazing with printing sintering support pillars

The support stress distribution of the printing sintering support pillars (PSSP), which became rounded due to the shrinkage that occurred in sintering formation, was different from that of the conventional cylindrical metal support pillars (CMSP). In this paper, ANSYS software was used to establish the mechanical simulation model of tempered vacuum glazing with PSSP. The influence of PSSP arrangement (spacing D, and three arrangement styles of square, regular triangle, and regular hexagon) on the mechanical properties of tempered vacuum glazing were analyzed, including the maximum stress σ1max on tempered glass (T-glass) surface, the maximum deformation ω1max of T-glass, and the maximum stress σ2max on the PSSP. The simulation results showed that when the PSSP were arranged in a square with D of 50 mm, σ1max, ω1max, and σ2max were 26.39 MPa, 7.82 μm, and 546.72 MPa, respectively, which met the requirements of the evaluation indexes of mechanical properties of tempered vacuum glazing, and had the least number of PSSP. The mechanical properties of tempered vacuum glazing with regular triangle arrangement of PSSP were better than those of square and regular hexagon arrangements, and the σ1max, ω1max and σ2max of which were 16.24 MPa, 7.25 μm and 489.66 MPa respectively, but with a too large number of PSSP. Optimization of the triangle layout angle and side length could reduce PSSP numbers. When the PSSP were arranged in an isosceles triangle with a 65° base angle and 50 mm bottom length, the mechanical properties of tempered vacuum glazing were the best. The triangular arrangement of PSSP had an optimization effect on the surface stress redistribution of tempered vacuum glazing, and the number of stress peaks was twice that of the square arrangement. Meanwhile, tempered vacuum glazing samples were made and the surface stress, thermal conductivity tests were conducted. The results showed that the measured values of peak and valley stress had an error of ±2.8 MPa compared to the simulated values. The simulation results were reliable and the surface stress redistribution of tempered vacuum glazing was present. The measured heat transfer coefficient of tempered vacuum glazing with PSSP was 0.7402 W m−2 K−1, slightly higher than that of tempered vacuum glazing with CMSP. The difference between them was not significant.

Revolutionising sustainability using a new triplet: A system dynamic model

The theory of sustainability has been analysed and implemented in various sectors to minimise the consumption of limited resources and to consume the fullest potential of existing resources. The triple-bottom concept of sustainability covers all possible interactions within an ecosystem. However, the dynamic nature and interconnectivity of sustainability systems, such as the environmental, economic, and social systems, are quite complex to curb sustainability challenges. The modification in one system may create disturbances in other systems. Based on the existing studies on sustainability, this study explored how these systems can be optimised by analysing the relationship between each side of sustainability triangle by 3As, i.e., Adaptability, Affordability and Availability, to determine their impacting macro and micro flows. The reinforcement can be achieved between social-environment by adaptability, social-economics by affordability and economics-environment by availability. These 3As synergise sustainability systems if inflows and outflows in sustainability systems are optimised. The system dynamic approach was adopted to model and examine all possible 3As’ inflows and outflows comprehensively in urban ecosystems. Micro flows are analysed for the associated macro flows for each factor of triplet. The results suggest that although sustainability systems are complex but not wicked in nature, it is required to focus on macro and micro flows of adaptability, affordability and availability in each system. This study may serve as a source of information for improving and maintaining sustainability in industries, businesses and the policy makers to optimise their existing resources on this comprehensive triplet model.

Air Flow Across Three Triangle Cylinders with Rounded Corners Edge in a Channel under Forced Convection

The analysis of heat transfer by forced convection steady, laminar flow around three triangle cylinders. A numerical investigation is conducted to analyze the variation in cross-section from a triangle to a circle by altering the edges of a triangular cylinder within a two-dimensional channel in the air. The effect of Reynolds number and transverse pitch (S_T) on the heat transfer characteristics are studied. Five Reynolds numbers (based on inlet height channel) are studied (Re = 100, 200, 400, 800, 1200) with a constant heat flux on cylinder surface (q^''= 5000 W/m^2), and S_T = 1.5D, 2D, 3D, the D is triangle side length. The governing partial differential equations, which include the continuity, momentum, and energy equations, are resolved by computational methods using workbench 2022R2 program. The results reveal that heat transfer rises as the transverse distance and the diameter of the rounded edge change.

Most expedient mine surveying analysis and geodetic method for tracking the stability of the tailings dam position

This article examines the issues of mine surveying and geodetic observations of the stability of the position of the tailings dam of the Almalyk Mining and Metallurgical Combine (AMMC) in Uzbekistan. For these purposes, a geodetic network was created on the principle of geodetic serifs, including the course of polygonometry of the 1st category. To determine the stability of the position and condition of the dam of the combined tailing dump in order to improve the accuracy, observations were made using satellite technologies and classical methods. From the work performed, it was concluded that when determining the coordinates of points, it is convenient to use the method of geodetic serifs. It allows simultaneous measurements of both the main course and lateral points, which significantly reduces time for making field measurements. In addition, in the case of short sides of triangles, the method of linear serifs is most appropriate, since it allows you to obtain more accurate values of the angles of triangles in comparison with angular measurements, which in the end result significantly increases the accuracy of determining the coordinates of points.

TO THE CALCULATION OF BANK PROTECTION MEASURES FOR CHANNELS SLOPES EXPOSED TO ALONGSHORE WAVES

Calculation methods for strengthening and covering slopes with paving stones or tiles have been developed, and precise formulas for calculating the diameters of paving stones have been adopted. The three most characteristic locations of the triangular base of the measuring pyramid on the slope are: a) When the stones forming the nest on the slope are positioned in such a way that the measuring stone placed on the tip of the pyramid is most easily overturned in the direction of the tidal current; b) When the measuring stone can be tilted in the direction of the shoreline, towards large widths; c) When the possible direction of overturning makes a large angle of 30° with the shoreline, that is when one of the sides of the equilateral triangle of the base of the pyramid is parallel to the shoreline.

The Postural and Body Surface Temperature Response of Leisure Horses to Lunging with Selected Lunging Aids.

Incorporating lunging into a horse's daily routine aims to enhance fitness, physical condition, and specific skills or exercises when using lunging aids (LAs). To assess the effectiveness of lunging, non-contact technologies like geometric morphometrics and infrared thermography can be employed. This study seeks to evaluate lunging efficiency based on the horse's posture and surface temperature when lunging with different head and neck positions. The study aims to determine if changes in a horse's posture correspond to increased metabolic activity, as indicated by body surface temperature. Thirteen horses included in the study were lunged with chambon (CH), rubber band (RB), and triangle side reins (TRs) as well as with a freely moving head (FMH). Images were taken in visible light and infrared. Principal Component Analysis (PCA) was used to analyze horse posture changes and a Pixel-Counting Protocol (PCP) was used to quantify surface temperature patterns. The horses' posture exhibited contrasting changes, reflected by a changing centroid shape (p < 0.0001) but not size (p > 0.05) when lunged with RB and TRs, but not CH. Different (p < 0.0001) surface temperature patterns were observed during lunging. FMH lunging resulted in lower temperatures over a larger surface, CH induced moderate temperatures on a smaller area, RB caused moderate to high temperatures across a broader surface, and TRs led to higher temperatures over a smaller region. The studied lunging cases returned different (p < 0.0001) surface temperature patterns. Lunging with FMH returned lower temperatures over a larger surface, CH moderate temperatures on a smaller area, RB moderate to high temperatures across a broader surface, and TRs higher temperatures over a smaller region. The proposed methods can be applied to evaluate the efficiency of lunging in horses.

New Ways to Calculate the Probability in the Bertrand Problem

We give two new ways of calculating the probability of a chord of circumference randomly selected being larger than the side of an equilateral triangle inscribed in the circumference (this problem is known as the Bertrand paradox). The first one employs an immersion in R4, and the second one uses a direct probability measure over the set of chords.

Spectral Analysis of the Adjacency Matrices for Alternating Quotients of Hyperbolic Triangle Group ▵*(3,q,r) for q &lt; r Primes

Hyperbolic triangle groups are found within the category of finitely generated groups. These are topological groups formed by the reflections along the sides of a hyperbolic triangle and acting properly discontinuously on the hyperbolic plane. Higman raised a question about the simplicity of finitely generated groups. The best known example of a simple group is the alternating group An, where n≥5. This article establishes a relation between the hyperbolic triangle group denoted as ▵*(3,7,r) and the alternating group. The approach involves employing coset diagrams to establish this connection. The construction of adjacency matrices for these coset diagrams is performed, followed by a detailed examination of their spectral characteristics.

Research on Multibeam Bathymetric System Based on Geometrical Relation Mo Model

In this paper, the basic principles of multibeam bathymetry system are discussed in depth, the development of which originates from the single-beam bathymetry technology. Through profound mathematical modelling and geometric relationship derivation, a systematic and detailed analysis is carried out for the coverage width of multibeam bathymetry and the overlap rate between two adjacent bands in the case that the survey line is parallel to the horizontal plane. Adopting the idea of combining numbers and shapes, combined with the triangle side angle relationship, we established a geometrical-mathematical model with an α-slope slant line, which lays a solid theoretical foundation for solving the problem. In this study, we successfully solved the expression of seawater depth D of the multibeam bathymetric system in the case that the direction of the survey line is parallel to the horizontal plane by the method of listing relations. At the same time, we make full use of the sine-cosine theorem of triangles to derive the coverage width of the bathymetric strip in depth. Combining these two organically, a complete and detailed expression for the coverage width is formed, which provides a powerful mathematical tool for the further study of deep-sea bathymetry technology. In addition, by applying the mathematical model to the vacant data in Table 1, we successfully fill in this missing information, demonstrating the feasibility and accuracy of the model in practical applications. This study not only makes remarkable progress in theory, but also provides strong support for practical applications in the field of ocean bathymetry.

Sex and age estimation utilizing the Mastoid, Bi-mastoid and Occipital triangles from three dimensional reconstructed models in subadults of a South African (KZN) Black population

The mastoid, bi-mastoid and occipital triangles at the base of the skull are known to be good indicators of sexual dimorphism. However, few studies assess age and sex in the subadult group. Therefore, the aim was to determine the age and sex of the human crania from a South African (KZN) Black population utilizing the triangles of the base of the skull of individuals between 0 and 18 years. The sample consisted of 64 computed tomography (CT) scans (32 male and 32 female) between 0 and 18 years of age. Three-dimensional reconstructed images and CT images were analysed using SLICER 3D. The following linear parameters were measured: the mastoidale - porion; porion - asterion; mastoidale - asterion; left mastoidale - opisthion; right mastoidale - opisthion; right mastoidale - left mastoidale; left asterion - lambda; right asterion - lambda; left asterion - right asterion. The study found that the mastoid, bi-mastoid and occipital triangles are good indicators for sexual dimorphism between the ages of 0–18 years. However, the mean values of the total area were higher in males than females. When comparing laterality, the right side of the mastoid triangle was more significant than the left (p-value=0.016), with the length of the mastoidale-porion being the only significant linear measurement. Results show that the linear measurement mastoidale-porion is highly correlated to age, with mean values of r = 0.87 and 0.88 for the left and right sides, respectively. Age is a confounding factor in sexual dimorphism. Additionally, the mastoid triangle is more accurate in age estimation.

Inequalities That Imply the Norm of a Linear Space Is Induced by an Inner Product

The aim of this paper is to investigate when a linear normed space is an inner product space. Several conditions in a linear normed space are formulated with the help of inequalities. Some of them are from the literature and others are new. We prove that these conditions are equivalent with the fact that the norm is induced by an inner product. One of the new results is the following: in an inner product space, the sum of opposite edges of a tetrahedron are the sides of an acute angled triangle. The converse of this result holds also. More precisely, this property characterizes inner product spaces. Another new result is the following: in a tetrahedron, the sum of squares of opposite edges are the lengths of a triangle. We prove also that this property characterizes inner product spaces. In addition, we give simpler proofs to some theorems already known from the publications of other authors.

Performance assessment of continuously operating reference stations using the deemed rover analysis approach with NRTK GNSS technique in Thailand

The study initiated an evaluation of the GNSS continuously operating reference stations (CORS) performance for Network-based Real-Time Kinematic (NRTK) positioning in Thailand, using the deemed rover analysis approach. Fifty CORS were selected as rovers with varying network geometries. Results indicated the highest performance, 79.9% of fixed solutions, for an average triangle side length of 54 km, while the lowest performance, 29.9%, was observed for a 93 km length with accuracies at a decimetre level. These initial explorations suggest that reducing side lengths enhances fixed solution efficiency and positioning accuracy at rovers. Occasional metre-level errors occurred, potentially linked to Internet instability.

Large-area WS2 Deposited on Sapphire and Its In-Plane Raman and PL Spectral Distributions

Large-area and high-quality WS2 monolayer has been fabricated on the Al2O3 substrate. Three typical WS2 configurations were adopted to examine the in-plane spectral properties. For the triangle WS2 monolayer, the PL light region exhibits a large peak wavelength and could be deduced to be the relaxation of the compressed strain of the WS2 monolayer and the low defect density. For triangle WS2 monolayer with multilayer WS2 on center, combining the peak intensity and position results of PL and Raman spectra, the line traces near the side of the center triangle can be demonstrated to be the defects or dislocations due to the exist of the central multilayer WS2. For large-area WS2 monolayer with crystal domain, PL area integrated mapping shows a uniform light region across the whole surface, except the existing dark crystal domain boundary. The dark line traces could be attributed to compressed strain in the WS2 monolayer due to the formation of the WS2/Al2O3 hybrid structure. The in-plane PL and Raman spectra and mapping exhibited in this work reveal the distribution of stress and defects in this system and further clarify the effects of stress and defects on the optoelectronic properties of WS2 monolayer on Al2O3.

Characterization of aperiodic surfaces with mesh-based parameters

For technical surfaces, it is important to know their functional purpose and to characterize them accordingly. Therefore, ISO 21920–2 in 2D and ISO 25178–2 in 3D offer parameters that can assess surface functional properties. The topographic portions of a surface, for example hills and dales, can be classified as features and evaluated using feature parameters. However, no parameter exists to describe the spatial distribution of features with regard to the degree of homogeneity for aperiodic surfaces. Here we show the application of the Delaunay triangulation to quantify the spatial distribution respectively the geometric relationship of features. Therefore, the feature points are determined by watershed analysis and the resulting point cloud is meshed in 2D. Based on that mean and standard deviation of the triangle side lengths and the area disorder (AD) are calculated as new parameters. The method is demonstrated for sandblasted and chrome-plated specimens. In addition simulation is used to generate more data for analysis. With the proposed approach the distinction and extent of uniform, homogeneous or inhomogeneous spatial distributions of features with parameter AD can be determined.