Geometry and Trigonometry

Abstract

In the chapter Geometry and trigonometry, the majority of problems can be treated either by classical proof techniques and properties provided in the regular school program or by using some additional proof techniques and theorems that are considered to be outside of the scope of the regular school program. The classical proof techniques, theorems, properties or mathematical laws used in the proofs included in this chapter are the properties of congruence of triangles, properties of similarities of triangles, the Pythagorean theorem, the properties of the right-angled triangle, law of sines and law of cosines, the properties of an area, the properties of inscribed and circumscribed quadrilaterals, trigonometric methods, the method of coordinates and applications of geometric inequalities (for more applications, see Sedrakyan and Sedrakyan, Geometric Inequalities, 2017, [1]). A large number of problems are related with the maxima and minima properties, such problems can be treated either by the different methods of estimations or by transformations and simplifications of the trigonometric expressions. In some cases, when we need to find the sum of the given trigonometric expressions, the main trick is to represent each of the summands as the difference of two expressions and then perform the corresponding simplifications in order to find the required sum. For the readers convenience, we list and provide the formulations of all used theorems and proof techniques that are outside of the scope of the regular school program.