Perfect Triangles : Rational points on Elliptic Curves

Abstract

A Heron triangle is a triangle that has three rational sides (a, b, c) and a rational area whereas a perfect triangle is a Heron triangle that has three rational medians (k, l, m). Finding a perfect triangle was stated as an open problem by Richard Guy in [14]. Numerous research have been done in the past to find such a triangle, unfortunately to date no one has ever found one, nor has proved its non-existence. However, on the bright side, there are partial results which shows that there exist triangles that satisfies five or even six of the seven parameters to be rational.Heron triangles with two rational medians are parametrized by the eight curves C1, · · · , C8 mentioned in [4], [5] and [1]. In this thesis, we devote our attention to study the curve C4 which has the property of satisfying conditions such that six of seven parameters given by three sides, two medians and area are rational. Our aim is to perform an extensive search to investigate if we can extract any perfect triangles from this curve. If it is impossible to generate perfect triangles from the curve C4, then we proof the core theorem of this thesis which is the non-existence of perfect triangle arising from this curve.The contribution of this work is to introduce a method that extends from [4] and [5] by intersecting equations parametrized by Heron triangle with two rational median with the condition of the last median to be rational.