New Theorem on Triangles-more Generalized Than Pythagoras Theorem

Abstract

This paper establishes a basic equation $a^n+b^n=c^n$ applicable for any triangle, having a, b and c as the sides with ‘c’ being the longest side and ‘n' is a number varying from 1 to infinity. Here, a, b, c and n need not always be integers. It also arrives at a relation between largest angle $\theta$ (opposite to the longest side ‘c’) and sides of the triangle with the equation based on cosine rule.The paper graphically and mathematically illustrates the relation between the angle $\theta$ and ‘n’, for different values of ‘n’ and ‘r’ (where ‘r' is the ratio of sides b/a) for the range of both ‘n' and ‘r' varying from 1 to infinity. The paper also shows that Pythagoras theorem is a particular case of the above fundamental equation, when $n=2$. The paper clearly illustrates with an example that the above fundamental equation is valid even when any one (or two or all) of the sides a, b or c will become non-integer values for all powers of $n > 2$. This gives a clear way of understanding the Fermat’s Last Theorem.