Angle Bisector Research Articles

A proof without words for angle bisectors of right triangles

In this article, we state an interesting geometric conservation property between the three angle bisectors of three similar right triangles and provide a proof without words for its justification. A GeoGebra applet is also presented to help with the understanding of the progression of the proof from inductive to deductive stage.

An Improved Hybrid Path Planning Algorithm in Indoor Environment

Abstract During the path planning of robots in the indoor unstructured complex environment, there are often problems such as unreachable target points, deflection in the planning process, and failure to avoid dynamic obstacles in time. To solve these problems, an improved hybrid indoor path planning algorithm was proposed, wherein the improved global path planning algorithm was effectually integrated with improved local path planning algorithm. Firstly, the heuristic factor of traditional A-Star algorithm was optimized, search range and nodes were reduced, and then the path generated by traditional A-Star algorithm for path planning was smoothed using the angle bisector tangent point method. Secondly, combining path and environment information, local path planning was undertaken by utilizing the improved artificial potential field algorithm, and the unreachable target points problem was addressed by adjusting the repulsive field parameters. Additionally, dynamic potential field function was constructed to make it have the ability to resolve dynamic obstacles. Finally, in the part of actual environment verification, a comparison was made in this paper to assess the performance of the traditional hybrid algorithm against the improved algorithm in terms of path planning. The consequences showed that, by the hybrid algorithm proposed in this paper, the path planning length was reduced by 10.3%, the running time was decreased by 12.5%, and 34 redundant nodes were eliminated. The consequences indicated that the hybrid algorithm can effectively address the indoor unstructured and complex path planning problems.

ABC-triangles

If we talk about the centre of a triangle, what might we be referring to? Any triangle has many different points that could regarded as its centre; in fact, Encyclopedia of Triangle Centres lists over 70 000 possibilities. Three of the most famous centres, that every triangle will possess (although they may coincide), are the incentre (where the three angle bisectors meet), the centroid (where the three medians meet) and the orthocentre (where the three altitudes meet). Proofs that these centres are well-defined and exist for every triangle are simple and satisfying, good examples of reasoning (if we are teachers) for our students. Proving the three altitudes of a triangle share a point using the scalar product of vectors is a wonderful demonstration of the power of this idea.

FROM EUCLID TO ARTIFICIAL INTELLIGENCE

We give 19 proofs of the famous Angle Bisector Theorem from Euclid's Elements. The first proof is the Euclid's original proof, the remaining proofs use the methods of Euclidean Geo\-metry, Trigonometry, Analytic Geometry, Complex Numbers, and Gr\"obner Bases. The Gr\"obner Bases proof is in the area of Automatic Proving and Artificial Intelligence, so that the proofs in a way symbolise the development of mathematics from 300 BC (the Euclid's time) to modern days. We discuss what the proofs illustrate and why they are important for Math Education. All the proofs, except the first one, are original.

THE BISECTOR TRIANGLE WITH A 90° ANGLE

For the examination of the triangle discussed in the article, profound knowledge of plane geometry is required, and grasping the specific approach to solving it can be quite challenging without prior knowledge of what to rely on. The focus is on a triangle with a 120° angle, commonly known as the Shebarshin problem. The article presents both computational and purely geometric methods for solving a problem of elevated complexity. This particular problem appears in entrance exams for universities and in mathematical Olympiads at the highest levels. The theme of singular triangles remains constantly relevant in geometry. For instance, many properties and theorems are known about an orthocentric triangle, which have been discussed in detail and already proven. I propose to consider in detail the properties and theorems of an equally interesting triangle, the sides of which connect the bases of the bisectors of the interior angles of the triangle. The goal of my article is to alleviate the drawback of complexity while retaining accessibility and originality in the geometric problem involving a triangle with a 120° angle and an inscribed bisectral triangle with a corresponding 90° angle.

On Directed Length Ratios in the Lorentz-Minkowski Plane

The linear structure of the Lorentz-Minkowski plane is almost the same as Euclidean plane. But, there is one different aspect. These planes have different distance functions. So, it can be interesting to study the Lorentz analogues of topics that include the distance concept in the Euclidean plane. Thus, in this study, we show that the relationship between Euclidean and Lorentz distances is given depending on the slope of the line segment. Following, we investigate Lorentz analogues of Thales’ theorem, Angle Bisector theorems, Menelaus’ theorem and Ceva’s theorem.

Inertial migration of spherical and oblate particles in a triangular microchannel.

The. inertial migration of both spherical and oblate particles within an equilateral triangular channel is studied numerically. Our study primarily focuses on the effects of fluid inertia, quantified by the Reynolds number (Re) and particle size (β). Our observations reveal two distinct equilibrium positions: the corner equilibrium position (CEP) is situated along the angle bisector near the corner, while the face equilibrium position (FEP) is located on a segment of the line perpendicular from the triangle's center to one of its sides. Spherical particles with varying initial positions predominantly reach the FEP. For oblate particles initially positioned along the angle bisector with a specific orientation, meaning the particle's evolution axis is inside the plane bisecting the angle, they will migrate along the angle bisector to reach the CEP while rotating in the tumbling mode. Conversely, for particles with different initial orientations and positions, they will employ the log-rolling mode to reach the FEP. Notably, we identify a dual-stage particle migration process to the FEP, with trajectories converging to an equilibrium manifold, which bears a resemblance to the cross sectionof the channel. To further illustrate the transition between FEP and CEP under general initial conditions, except for those along the angle bisector, we construct a phase diagram in the (Re, β) parameter space. This transition is often triggered by the size of larger particles (as the FEP cannot accommodate them) or the influence of inertia for smaller particles. For the FEP, especially for medium- or small-size particles, we notice an initial outward movement of the FEP from the center of the cross sectionas Re increases, followed by a return towards the center. This behavior results from the interplay of three forces acting on the particle. This research holds potential implications for the design of microfluidic devices, offering insights into the behavior of particles within equilateral triangular channels.

Proving concurrency by loci

A fascinating and catchy method for proving that a number of special lines concur is using the concept of locus. This is now the classical method for proving the concurrency of the internal angle bisectors and perpendicular side bisectors of a triangle. In this paper, we prove the concurrency of the altitudes and the medians by showing that they are loci of some interesting points. Our proofs for these ancient theorems seem to be new. We also provide loci method proofs for the concurrency theorems of Ceva and Carnot.

Validation of the Angle Bisector Method for Precise Tibiofibular Syndesmosis Fixation Angle Using Computed Tomography and 3D-Printed Models

Category: Ankle; Trauma Introduction/Purpose: The proposed ideal alignment for the syndesmosis involves a line connecting the centroids of the fibula and tibia, but there is uncertainty about the optimal intraoperative method for determining the appropriate fixation angle. This is due to the fact that the AO guidelines do not account for patient or level-specific factors, and the determination of the angle relative to the coronal plane relies on the surgeon's judgment. Inexperienced trauma surgeons may encounter significant problems with syndesmotic malalignment as a result. The goal of this study was to validate the effectiveness of the angle bisector method using computed tomography and 3D-printed ankle models, in order to determine whether it can assist in achieving proper screw trajectory for syndesmotic fixation. Methods: The angle bisector method was utilized to identify a precise trajectory for the true syndesmotic axis, using the bisector of the angle formed by two lines tangential to the anterior and posterior fibula and tibia. CT validation was made on CT angiography of 50 consecutive patients. DICOM data from 16 ankles were used to create 3D anatomical models which were printed using a desktop FDM printer. Two trauma surgeons performed syndesmotic fixations using the angle bisector method at 2 cm and 3.5 cm proximal to the joint space, after identifying the bisector of the angle formed by two K-wires tangential to the anterior and posterior fibula and tibia. The axial sections of CTs and printed 3D-models were analyzed using software to determine the relationship between centroidal axis and the inserted screws. The measurements were made twice by two blinded observers, with a two-week interval. Results: The results of the study show that the angle bisector method provides a reliable direction with minimal differences at both the 2 cm and 3.5 cm levels. The average angle between the centroidal axis and screw trajectory was 2.4° ± 2° at the 2 cm level and 1.3° ± 1.5° at the 3.5 cm level. Furthermore, the average distance between the fibular entry points of the centroidal axis and screw trajectory was less than 1 mm at both levels, indicating that the angle bisector method can serve as an excellent entry point for syndesmotic fixation on the fibula. The inter- and intra-surgeon analyses showed excellent consistency, with all ICC values above 0.90. Similarly, inter- and intra-observer consistencies regarding the measurements were also excellent. Conclusion: The study demonstrated that the angle bisector method was successful in identifying the original syndesmotic axis and aided in the precise placement of implants in the desired direction. The method has the potential to be replicated intraoperatively and could serve as a basis for the development of a novel surgical guide for determining the correct syndesmotic axis. However, safety analysis using cadavers should be conducted prior to clinical implementation.

Concurrent Measurement of Local and Global in-Plane Permeability of Reinforcement Fabrics through Real-Time Image Processing.

Properties of reinforcement fabrics, such as permeability, are typically characterized in a volume-averaging sense, whereas the fabric microstructure may vary spatially. This makes designing an effective resin infusion strategy for defect-free composite fabrication challenging. Our work presents a concurrent method for simultaneously measuring the local and global in-plane permeability and offers a handy technique for evaluating spatial variability. This experimental setup was similar to that of unidirectional in-plane permeability tests. The fabric, however, should be cut and tested along the angle bisector of warp and weft directions. The evolution of resin flow fronts was analyzed in real-time using in-house code through live video monitoring. The local and global in-plane permeability components were then obtained by applying Darcy's law regionally and globally. The results are in good agreement with those obtained by radial permeability experiments. Statistical analysis of local permeability reveals that the microstructure variability follows a normal distribution. A complete description of fabric microstructure provided by X-ray microcomputed tomography suggests that local permeability and microstructure variation are closely related, confirming the efficacy of the newly proposed method. This work enables the estimation of fabric structure variability and local and global in-plane permeability in a single test without resorting to expensive volume imaging techniques.

Algebraic Expressions for Three Sides of Triangles (Triangle Builders)

A triangle is a polygon with three edges and three vertices. There are formulas for areas,median,angle bisector,altitude etc. for triangles,but there are no algebraic expressions(with some exceptions)for three sides of various kinds of triangles.These are needed to build various kinds of triangles. I have given here all total seven types of algebraic expressions for three sides of various kinds of triangles.

Angle bisector method to determine the accurate angle for tibiofibular syndesmotic fixation: A validation study with 3D-printed anatomical models

PurposeThis study aimed to validate the angle bisector method on 3D-printed ankle models to reveal whether it aids in placing syndesmotic screws at an accurate trajectory that is patient- and level-specific and also not surgeon-dependent. MethodsDICOM data of 16 ankles were used to create 3D anatomical models. Then the models were printed in their original size and two trauma surgeons performed the syndesmotic fixations with the angle bisector method at 2 cm and 3.5 cm proximal to joint space. Afterward, the models were sectioned to reveal the trajectory of the screws. The photos of the axial sections were processed in a software to determine the centroidal axis which is defined as true syndesmotic axis and analyze its relationship with the screws inserted. The angle between the centroidal axis and syndesmotic screw was measured by two-blinded observers 2 times with 2 weeks interval. ResultsThe average angle between the centroidal axis and screw trajectory was 2.4° ± 2° at 2 cm-level and 1.3° ± 1.5° at 3.5 cm-level, indicating a reliable direction with minimal differences at both levels. The average distance between fibular entry points of the centroidal axis and screw trajectory was less than 1 mm at both levels indicating that the angle bisector method can provide an excellent entry point from fibula for syndesmotic fixation. The inter- & intra-observer consistencies were excellent with all ICC values above 0.90. ConclusionThe angle bisector method provided an accurate syndesmotic axis for implant placement which is patient- & level-specific and not surgeon-dependent, in 3D-printed anatomical ankle models.

Cortices of Fibula and Tibia Can Provide Landmarks for Accurate Syndesmosis Fixation Angle: Computed Tomography Validation of Angle Bisector Method

Anatomic syndesmosis reduction is necessary to restore ankle biomechanics and prevent poor clinical outcomes, but malreduction can be encountered frequently since the ideal fixation angle varies between patients and fixation levels. This study aimed to validate the angle bisector method to reveal whether it provides an accurate syndesmotic fixation angle that is patient- and level-specific. Lower extremity CT angiography of 50 consecutive patients (25 male, 25 female) without evident ankle pathology were evaluated. The average age was 52.8 (±18, range: 18-75). Lines tangent to anterior and posterior cortices of tibia and fibula were drawn in the axial plane at both 2 cm and 3.5 cm above the ankle joint line. Bisection of the angle formed between these lines was drawn and its relationship with the centroidal axis, which is proposed to be the ideal syndesmotic axis, was evaluated. The angle between the bisector line & the centroidal axis and the distance between their most lateral intersections with the fibula were calculated. The measurements were made by 3 blinded observers. Intra- and interobserver reliability analyses were conducted. The average centroidal axis-bisector angle was 2.1° ± 2.1° at 2 cm and 0.6° ± 1.3° at 3.5 cm level. The average distance to the actual syndesmosis entry point was 1.0 ± 0.9 mm at 2 cm and 0.4 ± 0.4 mm at 3.5 cm level. The values didn't show any significant difference according to gender. Intra- and interobserver reliability analysis showed excellent correlation in all parameters (interclass correlation coefficient > 0.90). Angle bisector method was found strongly reliable providing accurate direction for syndesmotic axis. It can provide a patient- and level-specific angle for the application of syndesmotic implants without increasing the fluoroscopy exposure. Its use can have a broad impact on functional outcomes of ankle injuries by decreasing the malreduction rates. Further cadaveric validation and safety studies should be conducted for possible clinical usage.

Theorems of Euclidean Geometry Through Calculus

We re-derive Thales, Pythagoras, Apollonius, Stewart, Heron, al Kashi, de Gua, Terquem, Ptolemy, Brahmagupta and Euler's theorems as well as the inscribed angle theorem, the law of sines, the circumradius, inradius and some angle bisector formulae, by assuming the existence of an unknown relation between the geometric quantities at stake, observing how the relation behaves under small deviations of those quantities, and naturally establishing differential equations that we integrate out. Applying the general solution to some specific situation gives a particular solution corresponding to the expected theorem. We also establish an equivalence between a polynomial equation and a set of partial differential equations. We finally comment on a differential equation which arises after a small scale transformation and should concern all relations between metric quantities.

On the Mastery of Elementary Geometric Concepts¤.

We publish here the report by N.F. Talyzina to a Conference on Psychology in 1955 and released two years later in the proceedings of that conference (N.F. Talyzina [1957]. In B.G. Anan′yeva et al. [Eds.], Materialy soveshchaniia po psikhologii. July 1–6, 1955. Moscow: Publishing House of the Academy of Pedagogical Sciences of the RSFSR.) This report is of particular interest, since it describes some of the first formative experiments based on the theory of planned stage-by-stage formation of mental actions; the text does not yet discuss the formation of actions, but only the application of the attributes of concepts to solving problems. The text presents the results of a study of the formation of elementary geometric concepts according to the theory of planned stage-by-stage formation of mental actions and concepts. Twenty-two subjects from grades 6–9 formed geometric concepts such as “line,” “angle,” “angle bisector,” “perpendicular,” “adjacent angles,” and “supplementary angles.” These attributes were formed in the process of their application to solving different types of problems. Observational and then formative (training) experiments were performed individually. The article contains excerpts from the subjects’ protocols as they solved the problems. It was shown that stage-by-stage development (identifying attributes from a definition, saying them aloud, and then applying them to solving problems) leads to mastery not only of the geometric concepts themselves, but also of the method of action with definitions in general, allowing students to transfer the method they have learned to concepts from another field of knowledge.

A new approach to the three bisectors problem

In 1994, Mironescu and Panaitopol proved the existence of a triangle with arbitrary prescribed angle bisector lengths. The proof is reduced to a particular fixed point problem. We generalize this problem and propose an alternative mathematical tool. A numerical example is provided. In this frame, we establish some local and uniform stability properties.

Uniqueness of the 2D Euler equation on a corner domain with non-constant vorticity around the corner

We consider the 2D incompressible Euler equation on a corner domain Ω with angle νπ with . We prove that if the initial vorticity ω 0 ∈ L 1(Ω) ∩ L ∞(Ω) and if ω 0 is non-negative and supported on one side of the angle bisector of the domain, then the weak solutions are unique. This is the first result which proves uniqueness when the velocity is far from Lipschitz and the initial vorticity is non-constant around the boundary.

Fuzzy Comprehensive Evaluation of Maritime Boundary Delimitation Schemes Based on AHP-Entropy Weight Method

The basic science region of marine management boundary lines among different provinces within the extent at least of the territorial sea is of great means to promote the sustainable growth of China's marine economy. There are 11 provinces along the coastal regions of the Chinese Mainland, and the scientific division of marine management boundaries among different provinces within the extents of the territorial sea is of great importance in promoting the sustainability of China's marine economy. We conducted a case study of maritime boundary demarcation between Jiangsu and Shandong Provinces, and a fuzzy evaluation index system with 14 indices in three subsets was developed to evaluate the benefits and drawbacks of maritime boundary delimitation, determine the optimal scheme, and address shortages in the current evaluation indicators and evaluation methods. The analytic hierarchy process (AHP) and the entropy weight method were used to calculate the combined weights of indices. Three maritime border delimitation schemes, namely, the historical boundary delimitation scheme, angle bisector delimitation scheme, and equidistance delimitation scheme, were evaluated using the comprehensive evaluation indices. Results show that the equidistance delimitation scheme is relatively superior to the two other schemes. The evaluation index is 0.504761, and the evaluation grade is “good.” The second best delimitation scheme is the angle bisector. The grade is “moderate,” and the evaluation index is 0.361641. The most ineffective boundary delimitation scheme is the historical one. The grade is “bad,” and the evaluation index is 0.135345. More consideration should be given to people's livelihoods and the safeguarding of national marine rights and interests in the late optimization of maritime boundary delimitation schemes. The fuzzy comprehensive evaluation technique based on AHP-entropy weight can help decision-makers choose the optimum scheme by providing a quantitative top-down sequence of schemes in terms of quality and solving estimate difficulties in maritime boundary delimitation schemes. As a result, it has a wide range of applications.

Harmonic Suppression and Size Reduction of a Low-Profile CP Patch Antenna by Using Nondegenerate Modes

A novel design method of harmonic suppression and size reduction of a circularly polarized (CP) microstrip patch antenna (MPA) is presented by using nondegenerate modes. Based on a circular MPA, a pair of orthogonal modes could be generated between these undesired resonant modes. Then, by loading the shorting pins in a quarter fan-shaped patch, all these undesirable high-order modes could be suppressed between them. After that, the higher and lower modes are moved closer to each other by cutting a long slot at the angle bisector of the fan-shaped patch, providing the CP radiation pattern. Finally, a prototype of this proposed antenna has been designed, produced, and measured to verify the design concept. The experimental results show that the proposed antenna has an impedance bandwidth of 286 MHz (13 774–14 060 MHz), axial ratio bandwidth of 80 MHz (13 900–13 980 MHz), and a left-hand CP peak gain of around 4.8 dBi. Most importantly, the CP antenna has realized the harmonic suppression of about 15 GHz, which has an obvious advantage as compared to the traditional CP MPA. Moreover, the antenna still holds the primary low-profile, single-radiator, simple structure, and high-efficiency simultaneously.

A triangular exploration

In this Article we study the following problem: Let ΔABC be an acute-angled triangle. Let the points D, E, F on the sides BC, CA and AB, respectively, be such that AD is the median from A, BE is the internal angle bisector of ∠ABC, and CF is the altitude from C. This is shown in Figure 1.