The application and promotion of ptolemy's theorem and inequalitiesin maximum-minimum value problems
托勒密定理及不等式在最值问题中的应用与推广

摘要

托勒密定理及托勒密不等式是平面几何中的重要定理及推论。运用托勒密定理及托勒密不等式不仅对解决平面圆内接四边形问题具有重要作用，还给出解决代数中某类最值问题的通用解法，这一解法拓展了代数问题几何化的解题思路，提供了快速解题的捷径。本文通过分析托勒密定理在一元双根式函数最大值问题中的解题思路，并举例说明托勒密不等式在平面几何一类最值问题中的解法，总结归纳出了代数中一元双根式函数最大值问题和平面几何中平面四边形某类最值问题的一般性结论。在代数与平面几何此类最值问题上，给中高考考生或参加数学竞赛等学有余力者创造出新的速解方法。

Abstract

Ptolemy 's theorem and Ptolemy 's inequality are important theorems and corollaries in plane geometry. The use of Ptolemy 's theorem and Ptolemy 's inequality not only plays an important role in solving the problem of plane circle inscribed quadrilateral, but also gives a general solution to a certain kind of maximum-minimum value problems in algebra. This solution expands the problem-solving idea of geometricization of algebraic problems and provides a shortcut for fast problem-solving. In this paper, by analyzing the solution of Ptolemy 's theorem in the maximum value problems of one-variable double-root function, and giving examples to illustrate the solution of Ptolemy 's inequality in the maximum-minimum value problem of plane geometry, the general conclusions of some kinds of maximum value problems of one-variable double-root function in algebra and the maximum-minimum value problems of plane quadrilateral in plane geometry are summarized. On the maximum-minimum value problems such as algebra and plane geometry, a new fast solution method is created for the candidates of high school entrance examination or those who have the ability to participate in mathematics competitions.