平面Bonnesen等周不等式的进一步加强

设欧氏平面R2中域D的面积为A,周长为L,r及R分别为D的最大内接圆半径及最小外接圆半径。利用参考文献中和分几何方法,给出了平面Bonnesen等周不等式的进一步加强,证明了L2-4πA≥π2(R-r)2(πR+πr-L)2.     

Inequalities for inscribed simplexes

The problem on the geometrc inequalities involving an n-dimensional simplex and its inscribed simplex is studied. Aninequality is established, which reveals that the difference between the squared circumradius of the n-dimensional simplex andthe squared distance between its circumcenter and barycenter times the squared circumradius of its inscribed simplex is not lessthan the 2(n-1)th power of n times its squared inradius, and is equal to when the simplex is regular and its inscribed siplex is atangent point one. Deduction from this inequality reaches a generalization of n-dimensional Euler inequality indicating that thecircumradius of the simplex is not less than the n-fold inradius. Another inequality is derived to present the relationship betweenthe circumradius of the n-dimensional simplex and the circumradius and inradius of its pedal simplex.     

Providing a Foundation for Deductive Reasoning: Students' Interpretations when Using Dynamic Geometry Software and Their Evolving Mathematical Explanations

A key issue for mathematics education is howchildren can be supported in shifting from `because it looks right' or`because it works in these cases' to convincing arguments which work ingeneral. In geometry, forms of software usually known as dynamicgeometry environments may be useful as they can enable students tointeract with geometrical theory. Yet the meanings that students gain ofdeductive reasoning through experience with such software is likely to beshaped, not only by the tasks they tackle and their interactions with theirteacher and with other students, but also by features of the softwareenvironment. In order to try to illuminate this latter phenomenon, and todetermine the longer-term influence of using such software, this paperreports on data from a longitudinal study of 12-year-old students'interpretations of geometrical objects and relationships when using dynamicgeometry software. The focus of the paper is the progressivemathematisation of the student's sense of the software, examining theirinterpretations and using the explanations that students give of thegeometrical properties of various quadrilaterals that they construct as oneindicator of this. The research suggests that the students' explanations canevolve from imprecise, `everyday' expressions, through reasoning that isovertly mediated by the software environment, to mathematicalexplanations of the geometric situation that transcend the particular toolbeing used. This latter stage, it is suggested, should help to provide afoundation on which to build further notions of deductive reasoning inmathematics.This revised version was published online in September 2005 with corrections to the Cover Date.