Problems Involving Triangles

Abstract

The first chapter is devoted to triangles and their properties. It covers the condition of existence of a triangle, trigonometric relationship in a right triangle, Laws of sines and cosines, Pythagorean Theorem, Thales’, Menelaus’, and Ceva’s Theorems and their applications. For example, in this chapter you will find out whether or not three given points are collinear. It is well known (but you will learn how to prove it) that three medians, bisectors or three altitudes concur. What if we take some points M, N, and K on the sides of a triangle and connect them with the corresponding opposite vertex? Under what condition will such cevians concur? Topics include similar triangles, areas of triangles, lemmas on the area of triangle, properties of a bisector, median, or height dropped from the same vertex.