Triangle Cevian and Side Relations for The Concurrent Case and The General Non-Concurrent Case

Abstract

I began this research study by first going through some existing work on triangle geometry and I came across some interesting theorems, namely Ceva’s theorem, Menelaus’ theorem, Steiner-Routh’s theorem and Van Aubel’s theorem. By studying the above theorems and through some friends I realized that I could develop a new approach of studying and analyzing the cevian and side segments of any triangle using a set of six linear equations that I have derived in this paper. The main contribution of this study is the proving Ceva's theorem and Menelaus' theorem, using a set of six equations derived using vectors. The equations are based on the proportions of the sides and cevians of a triangle and provide a unique and unconventional approach to solving problems in this field. One of the unique aspects of this approach is the use of vectors to derive the six equations. This paper presents the equations together with their derivations. I have shown how the six equations can be used as the basis of proving some famous triangle theorems. In addition to proving these existing theorems, I have also proven some relatively uncommon results in triangle geometry that can be useful for further research in this area. This therefore shows that these equations have the potential to reveal even deeper concepts on triangle Geometry that may have previously been unknown in triangle geometry.