Generalizations of Pythagoras Theorem to Polygons - Part 2

Abstract

The celebrated Greek mathematician Pythagoras, (circa 569 B.C.) re-discovered a legendary result, now known after him, as Pythagoras theorem:

Sum of squares of two mutually perpendicular sides in a right triangle equals the square of the hypotenuse, i.e. b² + p² = h², where b, p, h are the lengths of base, perpendicular and hypotenuse of the triangle. In the previous paper [3] several results were derived for quadrilaterals comprising of two right triangles expressing the left-hand expression, in above equation, as the sum of squares of 3 integers becoming the square of some fourth integer. The first fifteen Sections in [3] dealt with the direct sum of squares of some positive integers making the square of a fourth integer. Special choices for d = c + k, where k ranged over the set of natural numbers from 1 to 17 were discussed. Presently, we continue with the discussion for further integral powers of k = 18 onward.