Research For The Properties Of Multi-dimensional Angles And Inradius-circumradius For A Simplex

By using the theory and method of distance geometry and concex geometry, this thesis is concerned with the angles and inradius-circumradius porperties for a simplex, and obtained some metric properties of a simplex. The main content as follow.Chapter 1 introduces the concept of multi-dimensional angle and some concepts related, gets a sine law in another way for a simplex and obtains a new way to prove the second cosine law and the Bartos sine law for a simplex. In chapter 2 we focus our energy on how to use some different multi-dimensional angles to express a sort of angles which is constituted by two different dimensional planes, and get a kind of volume formulas. At the end of the chapter we get a new volume formula for a simplex and two theorams about the bisection planes of inner-outer dihedral angles of a simplex, in chapter 3 we consider the relations of the different multi-dimensional angles acquire a group angles inequality, from the group angles inequality may get some famous geometric inequalities. In chapter 4 we setup a geometric inequality about inradius-circumradius of a simplex from this inequality we can get some important geometric inequalities.