Advances in International Applied Mathematics

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.

Ptolemy s theorem; Ptolemaic inequality; Plane and geometry; The Maximum-minimum value problem; Application and promotion