• Communications of the Korean Mathematical Society
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• v.38 no.2
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• pp.561-573
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• 2023

In this paper, we extend the medial triangle theorem and Varignon's theorem to generic two-dimensional polygons and highlight the role played by diagonals in this process. One of the results is a synthetic definition of the concept of median for an n-sided polygon.

• The Mathematical Education
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• v.41 no.1
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• pp.79-90
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• 2002

In this article we study on connecting paper 131ding activities of triangle with mathematical proof Folding median, bisector of angle, and hight of paper triangle, we from and extract some ideas that help us to proof some important theorems of plane geometry. In this study using formed ideas in the process of paper folding activities, we suggest some interesting new mathematical proofs of the following theorems: 1. three medians of triangle are intersect in a point; 2. three bisectors of interior angles of triangle are intersect in a point; 3. three heights of triangle are intersect in a point.

• Journal of Integrative Natural Science
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• v.10 no.1
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• pp.27-32
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• 2017

In this paper we consider the two tangent lines of isogonal and isotomic conjugates of the line at both vertices of a given triangle. We find the necessary and sufficient condition for the two tangent lines of isogonal or isotomic conjugates of the line at both vertices and the median line to be concurrent. We also prove that every line whose isogonal conjugate tangent lines at both vertices are concurrent with the median line intersects at a unique point. Moreover, we show that the three intersection points correspond to the vertices of triangle are collinear.

• Anatomy and Cell Biology
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• v.56 no.2
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• pp.205-210
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• 2023

Left coronary artery divides into anterior interventricular branch and circumflex branch. As both the arteries run in their corresponding grooves, an arteriovenous trigone is formed between conus arteriosus and left auricle called triangle of Brocq and Mouchet. The triangle base is formed by great cardiac vein. This study aims to describe the frequency of triangle and its type and relationship between various boundaries and content of triangle and to supplement the existing knowledge of clinicians. This observational and descriptive study was conducted on 40 formalin fixed cadaveric hearts in department of anatomy, Kalpana chawla government medical college. The triangle was found in 92.5% of specimen with most common type being closed (51.3%) which is followed by inferiorly open in 35.1%, superiorly open in 8.1% and completely open in 5.4% hearts. Most frequent content of triangle was median artery followed by diagonal branches of anterior interventricular and circumflex branches. The mean area of the triangle was 246.3 mm². Relationship of vein with two arterial branches was either superficial or deep. The knowledge of different patterns of existence will be required for angiographic procedures. Further the triangle is a potential epicardial access route to left fibrous ring. Thus detailed knowledge of variations will help cardiologist to achieve better outcome in interventional procedures with minimal complications.

• School Mathematics
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• v.7 no.4
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• pp.391-402
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• 2005

The purpose of this study is to resolve misunderstanding for centroid of a triangle and to clarify several logical problems in finding the centroid of a Polygon. The conclusions are the followings. For a triangle, the misunderstanding that the centroid of a figure is the intersection of two lines that divide the area of the figure into two equal part is more easily accepted caused by the misinterpretation of a median. Concerning the equilibrium of a triangle, the median of it has the meaning that it makes the torques of both regions it divides to be equal, not the areas. The errors in students' strategies aiming for finding the centroid of a polygon fundamentally lie in the lack of their understanding of the mathematical investigation of physical phenomena. To investigate physical phenomena mathematically, we should abstract some mathematical principals from the phenomena which can provide the appropriate explanations for then. This abstraction is crucial because the development of mathematical theories for physical phenomena begins with those principals. However, the students weren't conscious of this process. Generally, we use the law of lever, the reciprocal proportionality of mass and distance, to explain the equilibrium of an object. But some self-evident principles in symmetry may also be logically sufficient to fix the centroid of a polygon. One of the studies by Archimedes, the famous ancient Greek mathematician, gives a solution to this rather awkward situation. He had developed the general theory of a centroid from a few axioms which concerns symmetry. But it should be noticed that these axioms are achieved from the abstraction of physical phenomena as well.

• Journal of Science Education
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• v.34 no.1
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• pp.175-184
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• 2010

We agree with Hanna and Jahnke's assertion on the use of arguments from physics in mathematical proofs and analyze their educational example of the use of arguments from physics in the proof of the center of gravity of a triangle. Moreover, we suggest practical models for the center of gravity of a triangle for the demonstration in a classroom. Comparing with the traditional mathematical arguments, the role of concepts and models from physics in arguments from physics will be clearly pointed out. Also, the necessity for arguments from physics in the classroom will be discussed in this paper.

• East Asian mathematical journal
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• v.29 no.4
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• pp.425-447
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• 2013

Since the center of mass of mathematics curriculum in middle school is dealt with only on triangle and it is defined as just an intersection point of median lines without any physical experiments, students sometimes have misconception of the centroid as well as it is difficult to promote divergent thinking that enables students to think the centroids of various figures. To overcome these problems and to instruct effectively the centroid unit in middle school mathematics classroom, this study suggests a teaching and learning method for the unit which uses physical experiments, drawing, and calculation methods sequentially based on the investigation of students' understanding on the centroid of triangle and the analysis of the mathematics textbooks.

• Communications of Mathematical Education
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• v.25 no.2
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• pp.381-402
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• 2011

In this paper we study formulae of the triangle's area. We solve problems related with making new formulae of the triangle's area. These formulae is consisted of some elements of triangle, for example side, angle, median, perimeter, radius of circumcircle. We transform formulae $S=\frac{1}{2}acsinB$, $S=\frac{abc}{4R}$, $S=\sqrt{p(p-a)(p-b)(p-c)}$, and make new formulae of the triangle's area. Some formulas are received in the process of Research and Education program in the science high school. We expect that our results will be used in the Research and Education program in the science high school.

• Journal of Institute of Control, Robotics and Systems
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• v.14 no.3
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• pp.236-241
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• 2008

This paper presents a method to recognize the plane regions for movement of walking robots. When the autonomous agencies using stereo camera or laser scanning sensor is under unknown 3D environment, the mobile agency has to detect the plane regions to decide the moving direction and perform the given tasks. In this paper, we propose a very fast method for plane detection using normal vector of a triangle by 3 vertices defined on a small circular region. To reduce the effect of noises and outliers, the triangle rotates with respect to the center position of the circular region and generates a series of triangles with different normal vectors based on different three points on the boundary of the circular region. The vectors for several triangles are normalized and then median direction of the normal vectors is used to test the planarity of the circular region. The method is very fast and we prove the performance of algorithm for real range data obtained from a stereo camera system.

• Communications of Mathematical Education
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• v.25 no.4
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• pp.681-691
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• 2011

Mathematics curriculum according to 2009 Revised National Curriculum suggests that school mathematics must cultivate interest and curiosity about mathematics in addition to creative thinking ability of students, and ability and attitude of observing and analyzing many things happening around. Centroid of a triangle in 2007 Revised National Curriculum is defined as 'an intersection point of three median lines of a triangle' and it has been instructed focusing on proof study that uses characteristic of parallel lines and similarity of a triangle. This could not teach by focusing on the centroid itself and there is a problem of planting a miss concept to students. And therefore this writing suggests centroid must be taught according to its essence that centroid is 'a dot that forms equilibrium', and a justification method about this could be different.