初等几何的起源、发展及启示

初等几何学大致经历了从经验几何发展到论证几何两个重要阶段。考察几何学的发展历程,从中可以获得一些有益的启示：初等几何学的教育价值在于通过学习初等几何学的基本内容,以掌握其所蕴含的思想方法,从而达到培养理性精神的目的。     

高等几何与初等几何的关系

高等几何是数学学科中的一门基础课程,其中有现代数学的思想和学习方法,它是初等几何的拓展,拓展了初等几何的解题方式,使初等几何的方法更加丰富,加深了初等几何的研究内容,延伸了初等几何的学习范围。高等几何有助于培养学生的几何基础,提高学生解决几何问题的能力,因此,学生应该加大对高等几何的重视程度,并认真研究高等几何在初等几何应用中的指导作用。文章主要是探讨高等几何和初等几何的关系。     

若干几何不等式的证明

运用Rrs法证明了[1][2]中关于三角形的三个几何不等式猜想。     

The effects of rigidity on school geometry learning

This study attempts to explain certain difficulties which ninth grade students face in tackling geometrical problems to prove, by relating them to general and to specific rigidity and cognitive style variables. The specific Geometrical Rigidity (GR) construct was conceived as comprising a perceptual component named Geometrical Functional Fixedness (GFF) and a conceptual component named Geometrical Method Embeddedness (GME). The general rigidity constructs were SDI and BRT that were derived within the Field Theory of K. Lewin and the Gestalt Theory respectively. The cognitive style construct was articulated-global style (measured by EFT). The results show that (a) GFF and GME are mutually independent (b) GR and its components have small negative correlations with SDI (c) GR and its components have insignificant correlations with BRT (d) GR and its components have strong negative correlations with articulated-global style (e) school geometry achievement has strong negative correlations with GR and its components, positive correlations with SDI and EFT, and insignificant correlations with BRT (f) GR is a potent and efficient predictor of future failure in school geometry learning. These results confirm the conceptual analysis of GR and indicate that GR has an independent existence as a cognitive style construct rather than a personality trait.     

面积和面积法

根据所提供的已知条件直接解题会使问题解决起来繁难，利用面积工具可得到比较简捷的解决．从面积计算与等积证明、利用等积变换处理平面几何问题、利用等积变换作图等展开论证．     

从蝴蝶定理的证明看高等几何对初等几何的作用

用高等几何方法证明了用初等几何方法较难证明的“蝴蝶定理” ,并给出定理的推广命题 ,从中显示出高等几何在初等几何中的作用     

几何最高点与物理最高点

在圆周运动中,经常遇到小球在竖直平面内做圆周运动的临界速度问题.小球能维持圆周运动的条件是能过最高点.而这里的最高点不一定是几何最高点,而应是物理最高点.几何最高点是图形中所画圆的最上端,是符合人眼视觉习惯的最高点.而物理最高点是物体在圆周运动过程中速度最小(称为临界速度)的点.物理最高点与几何最高点有时一致,有时不一致.     

对称变换法在解析几何中的应用

将对称变换法运用于求二次曲线的切线及二次曲面的切平面     

The teaching of geometry present problems and future aims

Summary Some general observations are made on present problems in the teaching of geometry at pre-college level, and some aims for the future are suggested. These remarks are inevitably made against a background of British experience. Aids for the teaching of geometry are discussed; and two lines of development, perspective drawing and nomograms, are described as a school introduction to geometrical ideas which are important at college.     

利用辛几何构作non-Cartesion认证码

利用辛几何构作了一个non-Cartesion认证码，计算了它的参数；当编码规则按等概率分布选取时，求出了模仿攻击和替换攻击成功的概率．     

Protective geometry

The following is a write-up of a talk that was presented at the TIFR as part of the Golden Jubilee celebrations of that Institute during 1996.     

从平面到空间推衍一组解析几何最小值问题

本文首先对平面解析几何中的一个最值问题进行研究，归纳出在平面解析几何该最值问题的一般性结论和性质，然后将该最值问题类推到空间，借鉴在平面解析几何该类型最值问题的研究方法，衍生出在空间解析几何类似最值问题的相应结论和性质。     

“代点法”在解析几何中的应用

从“代点法”的引入、变式、引中等几个方面，通过具体的实例探讨“代点法”的应用．同时，指出在应用“代点法”解决问题时，应该注意的问题．