一个新“九点圆”

在脍灸人口的《几何明珠》里,汪江松、黄家礼同志曾对三角形的九点圆作过一番精彩的描述和系统的论证。所谓三角形的九点圆,是指“任意三角形的三条高的垂足、三边的中点,以及垂心与顶点的三条连线的中点,这九点共圆”(见《几何明珠》P.138)。本文的目的是介绍一个由笔者发现的新的九点圆。叙述如下:如图1所示,△ABC为不     

四面体的十二点二次曲面

垂心四面体中四条高的垂足,四个面的重心及从各顶点与四面体的垂心连线的三等分点,共十二个点共球.试图把垂心改为四面体内的任意点,相应地把四条高线改换为过该点与每个顶点连线的共点直线组时,则将把垂心四面体的十二点球有趣地推广为四面体的十二点二次曲面.     

九点圆定理在三维空间的推广

众所周知,关于三角形有如下命题:九点圆定理在三角形中,以它的外心与垂心连线的中点为圆心,外接圆半径的一半为半径的圆,必通过9个特殊点,即:3个顶点与垂心连线的中点,3条边的中点,以及3条高的垂足.本文拟应用向量方法,将这个定理推广到三维空间的“共球有限点集”中.为此,我们     

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.     

与外周界中点三角形有关的不等式

文 [1]给出了三角形的周界中点的定义 :定义 1　如果三角形一边上的一点和这边所对的顶点把三角形的周界分为两条等长的折线 ,那么就称这一点为三角形的周界中点 .由于三角形任意两边之和大于第三边 ,因而三角形任一边上的周界中点必为这边的内点 .因此 ,我们不妨称定义 1中的周界中点为该三角形的内周界中点 ,以三个内周界中点为顶点的三角形称为该三角形的内周界中点三角形 .类似地 ,我们可以建立三角形的外周界中点及外周界中点三角形的概念 .定义 2　若将三角形的一条边延长 ,使其延长部分等于另两边之和 ,那么就称这条边与其延长部分构…     

圆内接闭折线垂心的两个性质

三角形有下面的性质[1](如图1):图1定理0设P是△ABC外接圆上弧BC的中点,Q是P的对径点,R是P关于边BC的对称点,H是△ABC的垂心,则AHRQ是平行四边形.这个性质是夫尔曼(Fuhrmann)发现的(三角形三顶点把外接圆分成三段弧的中点关于相应边的对称点所构成的三角形,被称为夫尔曼三角形)[1].本文将推广这个性质,证明圆内接闭折线的垂心的两个性质.为此,我们约定:符号A(n)表示平面内任意一条闭折线A1A2A3…AnA1.定理1设闭折线A(n)内接于⊙O,其垂心为H,Hjk是闭折线A(n)的2级顶点子集Vjk={A1,A2,…,Aj-1,Aj 1,…,Ak-1,Ak 1,…,An}的垂心…     

三角形的垂心的几个性质

文[1]介绍了三角形的垂心的如下性质:定理1三角形的垂心关于三边的对称点在这个三角形的外接圆上.本文以此为基础,提出如下性质:定理2三角形的垂心关于各边中点的对称点在三角形的外接圆上,且以这三个对称点为顶点的三角形与原三角形关于圆心中心对称.用符号语言表达即为:     

不等边三角形若干"心"的一个性质

笔者发现三角形“心”有如下性质:定理不等边三角形的内心I、垂心H、界心K及其旁心三角形的外心M是平行四边形的四个顶点.为了证明该定理,先给出如下几个引理:引理1△ABC中AD、BE、CF为三边上的高,垂心为H,则该三角形三边之中点,三个垂足D、E、F,三线段H A、H B、H C之中点九点     

垂心组的性质及应用

由三角形的三顶点及垂心引发我们给出垂心组的概念：以三点为三角形的顶点,另一点为该三角形的垂心的四点称为垂心组．由此即知,垂心组中的四点,每一点都可为其余三点为顶点的三角形的垂心;还可推知,垂心组有如下的优美性质．     

从三角形的垂心谈起--向量方法的一个应用

本文将三角形的垂心概念推广到圆内接四边形和圆内接五边形当中去 ,并且同时给出关于垂心的一条重要性质 .本文主要应用向量方法 .首先给出两条简易的引理 ,本文不加证明 .引理 1　设M是线段AB的中点 ,O为任意一点 ,则有OM =12 (OA+ OB) .引理 2　设G是△ABC的重心 ,O为任意一点 (在或不在△ABC所决定的平面上 ) ,则有OG=13(OA+ OB+ OC) .现在从三角形的垂心谈起 .图 1设O是△ABC的外心 ,OP⊥BC ,P是BC的中点 ,AQ是BC边上的高 (图 1 ) .在高AQ所在直线上取一点H ,使AH =2 OP ,则有OH =OA +AH=OA + 2OP=OA+ OB+ OC…     

Orthocentric simplices and their centers

A simplex is said to be orthocentric if its altitudes intersect in a common point, called its orthocenter. In this paper it is proved that if any two of the traditional centers of an orthocentric simplex (in any dimension) coincide, then the simplex is regular. Along the way orthocentric simplices in which all facets have the same circumradius are characterized, and the possible barycentric coordinates of the orthocenter are described precisely. In particular these barycentric coordinates are used to parametrize the shapes of orthocentric simplices. The substantial, but widespread, literature on orthocentric simplices is briefly surveyed in order to place the new results in their proper context, and some of the previously known results are given with new proofs from the present perspective.     

Concurrency of the altitudes of a triangle

Of all the traditional (or Greek) centers of a triangle, the orthocenter (i.e., the point of concurrence of the altitudes) is probably the one that attracted the most of attention. This may be due to the fact that it is the only one that has no exact analogue for arbitrary higher dimensional simplices, for spherical and hyperbolic triangles, or for triangles in normed planes. But it possibly has to do also with the non-existence of any explicit treatment of this center in the Greek works that have come down to us. In this paper we present different proofs of the fact that the altitudes of a triangle are concurrent. These include the first extant proof, in the works of al-Kūhī, Newton’s proof, Gauss’s proof, and other interesting proofs.     

The construction of a special equilateral triangle

This article deals with the construction of an equilateral triangle that must satisfy the following special constraint conditions. If the equilateral triangle is denoted by ΔABC, then the radii of the inscribed circle, the three escribed circles of ΔABC, and the circumcircle of ΔABC all must have positive integral radii. The inscribed circle radius is required to be 1 unit. The three escribed circles that have equal radii must have 3 units each, and the circumcircle of the triangle must have 2 units. All these requirements may seem outlandish. The aim is to teach crucial Geometric principles that Geometric designs must take into account before the constructions are implemented. This article hopefully may be useful to students of College Geometry as well as teachers.     

Remarks on orthocenters,Pappus’ theorem and Butterfly theorems

We present a generalization of the notion of the orthocenter of a triangle and of Pappus’ theorem. Both subjects were discussed with Pickert in the last year of his life. Furthermore we add a projective Butterfly theorem which covers all known affine cases.     

An unexpected appearance of Steiner's hypocycloid

J. de Cicco [1939] observed that two parabolas must touch each other if they have parallel axes, while one parabola touches the three sides of a given triangle and the other passes through the midpoints of those sides. Coxeter [1983] showed that the locus of the point of contact of the two parabolas, if the triangle is kept fixed while the common axial direction varies, is a rational cubic curve. In a subsequent paper, Coxeter investigated other aspects of this cubic (Coxeter [1985]).De Cicco's theorem, viewed as a result in the projective plane, can be dualized in a natural way. The cubic then becomes a set of lines, enveloping a curve of class three. We shall show that this curve is a quartic curve with three cusps, which is projectively equivalent to Steiner's well-known hypocycloid.     

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.     

Generalization of the triangle and Ptolemy inequalities

The Euclidean triangle inequality generalizes to an alternating inequality for any oddsided polygon that can be inscribed in a circle; there is equality in the even cases. A generalization of Ptolemy's theorem follows by inversion. The results have Minkowskian analogues.     

Teaching proof in the context of physics

This paper describes an application of statics to geometrical proofs in the classroom. The aim of the study was to find out whether the use of concepts and arguments from statics can help students understand and produce proofs of geometrical theorems. The two theorems studied were (1) that the medians in a triangle meet at a single point which is the centre of gravity of the triangle, and (2) the Varignon theorem, that the lines joining the midpoints of successive sides of a quadrilateral form a parallelogram. The classroom experiment showed that most students were successful in using arguments from statics in their proofs, and that they gained a better understanding of the theorems. These findings lend support to the claim that the introduction of statics helps students produce proofs and grasp their meaning.