Algebraic applications to combinatorial geometry

Combinatorial geometry is a branch of mathematics that combines principles from both combinatorics and geometry to study geometric structures and arrangements. In combinatorial geometry, the emphasis is on studying geometric objects and their relationships using combinatorial and algebraic techniques, such as counting, enumeration, and combinatorial optimization. The objects of interest may include points, lines, polygons, polyhedra, graphs, and other discrete structures. Combinatorial geometry finds applications in various areas, including computer science, optimization, and algorithm design. A famous problem in combinatorial geometry is called the Sylvester-Gallai problem. The Sylvester-Gallai problem, named after James Joseph Sylvester and Tibor Gallai. It asks whether, given a finite set of non- collinear points in the plane, there always exists a line that passes through exactly two of those points. This problem has been the subject of investigation for many years and has generated significant interest in combinatorial geometry. It was first posed by Sylvester in the 19th century and later solved by Gallai in 1944. The Sylvester-Gallai theorem, states that the answer to the problem is affirmative; that is, for any finite set of non-collinear points in the plane, there always exists a line that passes through exactly two of those points. It continues to be an active area of research. In this project we intend to study variations of this problem. The problem of constructing a triangle with rational geometric parameters is known as the "rational triangle problem." Specifically, it asks whether it is possible to construct a triangle whose side lengths, heights, medians, and area are all rational numbers. Where by median we mean a line segment joining a vertex to the midpoint of the opposite side. This problem was asked as an open problem by Richard Guy. The problem of constructing rational triangle is related to Diophantine equations, which seek integer solutions to polynomial equations. Despite many attempts, a general solution to the problem remains elusive. In this project we are planning to investigate this problem and a variant of this problem.