The purpose of this thesis is to study the polytopes in Eucliaeam spaces. Triangulations of polytopes are discussed brie y. Faces of ploytopes, such as vertices, edges, gridges. facets and so on, are de ned by their extremal subsets. Based on linear combinations and inner products of vectors., every concept we need is de ned carefully in order to give complete proof of the theorems. The work may be regard as a function for studing the theory of sperner's lemma on triangulation of polytopes.