英语高手来，帮忙把中文翻译成英文！一定要准确无误！谢谢了！

Average value inequality is at the core position in the theory of inequality, is one of the most important basic inequalities in mathematics, therefore, in many fields of mathematics have broad application. This paper simply introduces the method to prove inequality of several representative, including the backward induction,method, probability inequalities proof, and summarizes its application in various aspects of function and equation, analytic geometry, plane geometry, solid geometry inequality, triangle inequality problem and the triangle inequality,extremum problems, sequence and limit. Provides convenience for future study ofthis problem, help to solve other inequalities proof.

那个……因为那些关键字我不太懂怎么翻译，所以你可能需要改一下（我都用“”括出来了）。自认翻译做的不咋地，但还勉强拿得出手吧。

“Inequality of arithmetic and geometric means“ is the core of "inequality theory". It is one of the most important basic "inequalities" in mathematics. Thus, it is widely used in many mathematics areas. This "thesis /study /article /book /paper" gives a brief introduction on several representative methods to prove the "inequality theory", including “backward induction, probabilistic methods, inequality proof etc”, and summarized the application of "inequality theory" in areas such as "equations and functions, analytic geometry, plane geometry inequality, three-dimensional geometry inequality, triangle and the triangle inequality title card, extreme problem, limit the number of columns and various of other areas". Providing convenience for future studies in these areas, and helped in addressing the proof of other "ineuquality".

Keywords: "mean inequality; equations and functions; analytic geometry; geometric inequalities; extreme problems limit the number of columns"

Mean inequality is the core of inequality theory as well as one of the basic mathematics of the most important. So it is applied widely in many areas of mathematics. In this artical, some representice methods of the average inequality proof problems are introduced, including backward induction, probabilistic methods, inequality proof, etc. Their applications in equations and functions, analytic geometry, plane geometry inequality, inequality solid geometry, trigonometry certificate title, the triangle inequality, extreme problems and limit of a sequence etc. are also sumarized. This would make the future research on such simialr questions more conveinent as well as help solve the other problems of inequality proof.