托勒密定理在高中数学中的应用

The Application of Ptolemy’s Theorem in High School Mathematics

摘要: 自《普通高中数学课程标准(2017年版2020年修订)》颁布以来，数学课程的改革不断地深入，这对教师的数学专业知识素养提出了更高的要求，需要教师站在高观点视角下去理解中学数学知识的本质。除此之外，高考中频繁出现的具有高等数学背景的试题也对学生的认知提出了更高的要求。托勒密定理描述了圆内接四边形的四边与对角线之间的数量关系，在圆的几何学中起着独特的作用，利用它可以解决与圆有关的几何问题，也可以构造特殊的圆内接四边形来解决代数问题，将代数问题几何化，则很多问题的解决将更加方便且简单。通过对托勒密定理在高中数学中的多方面应用的研究，本文旨在激发学生对数学的兴趣，提高他们对数学概念的理解水平，为教师提供有效的教学工具，使学生更深入地理解数学知识，并在实际问题中灵活运用所学概念。

Abstract: Since the promulgation of the “General High School Mathematics Curriculum Standards (2017 Edition, 2020 Revision)”, the reform of mathematics curriculum has been continuously deepened, which has put forward higher requirements for teachers’ mathematical professional knowledge and literacy. Teachers need to understand the essence of middle school mathematics knowledge from a high perspective. In addition, the frequent appearance of questions with a background in advanced mathematics in the college entrance examination also puts higher demands on students’ cognition. Ptolemy’s theorem describes the quantitative relationship between the four sides and diagonals of a circle inscribed with a quadrilateral, playing a unique role in the geometry of circles. It can be used to solve geometric problems related to circles, as well as to construct special circles inscribed with quadrilaterals to solve algebraic problems. By geometricizing algebraic problems, many problems can be solved more conveniently and simply. Through the study of the various applications of Ptolemy’s theorem in high school mathematics, this article aims to stimulate students’ interest in mathematics, improve their understanding of mathematical concepts, provide effective teaching tools for teachers, enable students to have a deeper understanding of mathematical knowledge, and flexibly apply learned concepts in practical problems.