The subject matter of this article is a beautiful theorem developed by Ptolemy (about 100-178) in Chapters 10 and 11 of the first book of Almagest, the Great Collection of Astronomy in 13 books. In order to solve astronomical problems, Ptolemy used mathematical tools to calculate the length of the chord in the circle of radius $60$ as a function of the central angle. The application of Ptolemy's theorem is presented by giving some interesting examples. The paper analyzes the results in solving problems from the competitions in which Ptolemy's theorem is applied.