A Study On Vertex 's Angle Triangle

The Analytic Art is an important algebraic work of Francois Vieta. This book not only contains the symbol system, algebraic equations and other important contents, but also includes the research of diving angle. The problem of diving an angle equally is one of the concerns for mathematicians. This problem comes from trisection of an angle, which is one of the three geometric problems in ancient Greek. This paper based on the background of the history of mathematics and to review the related literature reproduces Vieta solved the problem of diving an arbitrary angle with multiple-angle triangle as the breakthrough point.The main work of this paper as follows:Firstly, the paper analyses the study in multiple-angle triangle by predecessor and introduces the life of Francois Vieta and the book of The Analytic Art, which is based on the translation of The Analytic Art and consulted the research literature.Secondly, the paper researches the construction method of synaeresic triangle and diaeresic triangle and traces synaeresic triangle and diaeresic triangle coming from sine and cosine functions of the sum and difference of two angles. This paper also shows Vieta constructed the multiple-angle triangle by the method of synaeresic triangle. It summarizes the sine and cosine formulas by the the relationship between the characteristics of three sides in multiple-angle triangles and the items of binomial expansion. It also expresses the bases of multiple-angle triangles as higher-degree algebraic expression of one unknown by using the proportionals in continued proportion.Thirdly, this paper studies the method to solve 45-degree equation by the sine and cosine identical equation of Francois Vieta and provides one way to work out the higher-degree equations of this kind. It also researches Vieta transferred the problem of dividing an arbitrary angle equally into solving the numerical equation of corresponding degree with the help of multiple-angle triangle. Consequently, this paper completes dividing an arbitrary angle equally.At last, this part summarizes all the research in this paper and list the problems to be solved.