Greetings fellow math majors and professors of Canisius College. For three years now I have been conducting independent research in the the field of Euclidean Geometry. Specifically, Applications of Euclidean Postulate 9: The Plane Separation Postulate. The purpose of my research was to develop a theorem that tells the number of disjoint subsets there are in a given set, when lines are drawn to connect two different boundary points. After many unsuccessful attempts at creating such a theorem, I do believe that I have finally developed such a theorem. Based on the number of straight lines and lines involved in each cross, the number of disjoint subsets can be calculated. Although the foundation of this theorem is geometrical, I will also show how this theorem can help solve simple algebra problems. I will also be presenting a blueprint on how this theorem may be adapted to curves if further research is to be conducted.
Theorem: Let Q be a Jordan curve along with its interior. The number of disjoint subsets inside the Jordan curve is represented by S=(n+1)+sum(gamma_p_i -1) where Q is cut by n lines which can intersect at a finite number of points $p_1, p_2,…,p_K so that gamma_p_i denotes the number of lines meeting at point p_i.
Hope you guys enjoyed my presentation and learned quite a bit about euclidean geometry, topology and possibly what it takes to one day create your own theorem.