A way of proving colinear relationship using ratio properties of parallel lines and other theorems

Abstract

Plane geometry has long been an essential part in Mathematics since the subject was born. Though it is no longer appealing to most researchers, plane geometry problems are now used in mathematics Olympics to test students’ ability and countless minds are still exploring the beauty and possibilities created by two-dimension geometries. The author independently discovered a useful method of solving geometric problems and has successfully applied it on three quite different problems that had appeared in mathematics Olympics, which indicate that the method is not a coincidence or a flashy skill, but a sharp observation of a common configuration: parallel. Below, the method will be introduced more comprehensively, and some deeper understanding of it will also be revealed.

       First, the discovery of the method will be briefly explained in a solution. Then, two applications will be stated demonstrating broader uses of the method and rich properties related. Eventually, the pattern of intersecting lines and parallel lines will be further described with several relating theorems.