The characterizations of triangles using the nine-point circles

In this note, primarily intended for high school students and high school teachers, characterizations of a right triangle and an equilateral triangle in the Euclidean plane are presented using the nine-point circle of a given triangle. Geometrical applications are explored along with their possible uses in the teaching environment.     

三角形的九点二次曲线

在任意三角形内,三边中点,三高的垂足,以及连接顶点与垂心的三线段的中点,都在同一圆上,此圆即为三角形九点圆.三角形的九点圆是欧氏几何中著名的优美定理,被称为欧拉圆和费尔巴哈圆.本文试图把垂心改换为平面内的任意点,相应地把三条高线改换为过每个顶点各一条的共点直线组时,则将把三角形的九点圆有趣地推广为三角形的九点二次曲线.并具体讨论在不同的区域内得到的九点二次曲线.     

Classroom note: A Generalization of Leibniz rule for higher derivatives

This note provides a simple method to extend the usual Leibniz rule for higher derivatives of the product of two functions to several functions, which is within the reach of freshman calculus students.     

Relations among five radii of circles in a triangle,its sides and other segments

Students use GeoGebra to explore the mathematical relations among different radii of circles in a triangle (circumcircle, incircle, excircles) and the sides and other segments in the triangle. The more formal mathematical development of the relations that follows the explorations is based on known geometrical properties, different formulas relating the radii to the sides and the inequalities between the different averages. The activities described were conducted with pre-service teachers of mathematics, with empirical investigation of the relations using dynamical geometry software, and formal presentation of proofs.     

An enumeration of equilateral triangle dissections

We enumerate all dissections of an equilateral triangle into smaller equilateral triangles up to size 20, where each triangle has integer side lengths. A perfect dissection has no two triangles of the same side, counting up- and down-oriented triangles as different. We computationally prove Tutte’s conjecture that the smallest perfect dissection has size 15 and we find all perfect dissections up to size 20.     

A sacred geometry of the equilateral triangle

In this article, we investigate the construction of spirals on an equilateral triangle and prove that these spirals are geometric. In further analysing these spirals we show that both the male (straight line segments) and female (curves) forms of the spiral exhibit exactly the same growth ratios and that these growth ratios are constant independent of the iteration of the spiral. In particular, we show that ratio of any two successive radius vectors from the ‘centre’ of the spiral as we move inwards towards that ‘centre’ is always 1/2. This same elegant result is also shown to be true for successive chords. All our results are demonstrated using mostly coordinate and transformational geometry. Finally we look at two methods for constructing these spirals with ruler and compass to maximum accuracy.     

Distances of group tables and latin squares via equilateral triangle dissections

Denote by gdist(p)$\mathrm{gdist}(p)$ the least non-zero number of cells that have to be changed to get a latin square from the table of addition modulo p  . A conjecture of Drápal, Cavenagh and Wanless states that there exists c>0$c>0$ such that gdist(p)?clog(p)$\mathrm{gdist}(p)?c\log (p)$. In this paper the conjecture is proved for c≈7.21$c\approx 7.21$, and as an intermediate result it is shown that an equilateral triangle of side n   can be non-trivially dissected into at most 5_log2(n)$5{\log }_2(n)$ integer-sided equilateral triangles. The paper also presents some evidence which suggests that gdist(p)/log(p)≈3.56$\mathrm{gdist}(p)/\log (p)\approx 3.56$ for large values of p.     

Eigenvalues and eigenfunctions of the Laplace operator on an equilateral triangle

A boundary value problem for the Laplace equation with Dirichlet and Neumann boundary conditions on an equilateral triangle is transformed to a problem of the same type on a rectangle. This enables us to use, e.g., the cyclic reduction method for computing the numerical solution of the problem. By the same transformation, explicit formulae for all eigenvalues and all eigenfunctions of the corresponding operator are obtained.     

Dynamic investigation of triangles inscribed in a circle,which tend to an equilateral triangle

We present a geometrical investigation of the process of creating an infinite sequence of triangles inscribed in a circle, whose areas, perimeters and lengths of radii of the inscribed circles tend to a limit in a monotonous manner.

First, using geometrical software, we investigate four theorems that represent interesting geometrical properties, after which we present formal proofs that rest on a combination between different fields of mathematics: trigonometry, algebra and geometry, and the use of the concept of standard deviation that is taken from statistics.     

Parallel packing and covering of an equilateral triangle with sequences of squares

An equilateral triangle T _e of sides 1 can be parallel covered with any sequence of squares whose total area is not smaller than 1:5. Moreover, any sequence of squares whose total area does not exceed $ \frac{3} {4}(2 - \sqrt 3 ) $ \frac{3} {4}(2 - \sqrt 3 ) (2 − √3) can be parallel packed into T _e .     

The use of digital technology in finding multiple paths to solve and extend an equilateral triangle task

Mathematical tasks are crucial elements for teachers to orient, foster and assess students’ processes to comprehend and develop mathematical knowledge. During the process of working and solving a task, searching for or discussing multiple solution paths becomes a powerful strategy for students to engage in mathematical thinking. A simple task that involves the construction of an equilateral triangle is used to present and discuss multiple solution approaches that rely on a variety of concepts and ways of reasoning. To this end, the use of a Dynamic Geometry System (GeoGebra) became instrumental in constructing and exploring dynamic models of the task. These model explorations provided a means to generate novel mathematical results.     

利用三角形及其九点圆的摄像机标定

提出基于三角形及其九点圆的摄像机标定方法.利用了三角形九点圆中其九个点的特殊性,并且利用透视投影变换保二次曲线不变性,得到其像点在像平面共椭圆,从而可以通过九点的映射关系将透视投影变换的非线性问题线性化.图像分割和角点提取的误差会直接影响标定的精度,在此三角形及其九点圆中的点特别是算法中的关键点三角形顶点和垂心都是三条直线的交点,减小图像分割与提取时造成的误差.DLT方法的不精确就源于图像分割和角点提取的误差,方法克服了DLT方法的不足.张的方法无法保证单应性矩阵的正交性,因此为了保证正交性和提高精度需要优化.与传统方法相比操作简单,应用九点圆定理,仿射变换的引入将透视投影非线性问题线性化,避免了参数之间的非线性方程求解,降低了参数求解的复杂性,因此其定标过程快捷,准确.模板的构造,减少了图像分割和交点提取误差,算法实现保证旋转矩阵的正交性.综合上述分析,理论上表明方法的有效性.同时实验表明,标定方法操作简单,不需要计算机视觉的专业知识,快速,精度高,鲁棒性好.     

The Magnus embedding as a special case of a homological construction

Let E be a group extension with Abelian kernel. Then it can be assigned an extension E′ of modules over the group ring of the quotient group. As a consequence, an embedding of the initial extension in some splitting extension arises. We prove that the celebrated Magnus embedding is a special case of this general construction.     

The dissection of a polygon into nearly equilateral triangles

Every polygon can be dissected into acute triangles. In this paper we prove that every polygon, such that the interior angles are at least /5, can be dissected into triangles with interior angles all less than or equal to 2/5. We find necessary conditions on the interior angles of the polygon in order to obtain a dissection into triangles with interior angles all (where /3<<2/5). The conjecture can be stated that these conditions are also sufficient.     

On the complexity of approximating the maximal inscribed ellipsoid for a polytope

We give a new polynomial bound on the complexity of approximating the maximal inscribed ellipsoid for a polytope.Research supported by NSF Grant DMS-8706133.Research supported by NSF Grant DMS-8904406.     

The trigonometric triangle

A procedure is presented by which the integration of a function such as cos ⁿ (x) or sin ⁿ (x) with n = 1, 2, 3, 4 … is facilitated.     

Number-theoretic interpretation and construction of a digital circle

This paper presents a new interpretation of a digital circle in terms of the distribution of square numbers in discrete intervals. The number-theoretic analysis that leads to many important properties of a digital circle succinctly captures the original perspectives of digital calculus and digital geometry for its visualization and characterization. To demonstrate the capability and efficacy of the proposed method, two simple algorithms for the construction of digital circles, based on simple number-theoretic concepts, have been reported. Both the algorithms require only a few primitive operations and are completely devoid of any floating-point computation. To speed up the computation, especially for circular arcs of high radii, a hybridized version of these two algorithms has been given. Experimental results have been furnished to elucidate the analytical power and algorithmic efficiency of the proposed approach. It has been also shown, how and why, for sufficiently high radius, the number-theoretic technique can expedite a circle construction algorithm.