Some Inequalities in a Triangle in Which the Length of One Side and the Inradius Are Given

In the article, we prove 18 inequalities involving inradius, a length of one side and one additional element of a given triangle. 14 of these inequalities are the necessary and sufficient conditions for the existence and uniqueness of such a triangle. All proofs are based on standard methods of calculus and can serve as a good demonstration of the relationship between different branches of mathematics (geometry, algebra, trigonometry, calculus). The article can be used by teachers and students in courses on advanced classical geometry.