Pythagorean Quadrilaterals

The Pythagorean Theorem says that if a and b are the leg lengths of a right triangle with hypotenuse c, then a 2 + b 2 = c 2. The infinitely many integral solutions are well classified, and numerous generalizations have been thoroughly studied. Lagrange [6] proved that every positive integer was the sum of 4 squares of integers. Waring [9] conjectured and Hilbert [5] proved that for every positive integer n there was a constant c(n) such that every positive integers was the sum of c(n) non-negative nth powers. Fermat, Euler, Gauss and Jacobi studied the number of solutions to a 2 + b 2 = n for fixed n (see eg. [3]). Lucas [7] challenged his readers to find all solutions n and c to $$ \sum\nolimits_{i = 1}^n {{i^2}} = {c^2} $$ , (for a nice solution see [1]) and Pell’s equation a 2 - kb 2 = ±1 for fixed k occupied the attention of many mathematicians. And finally there is the problem posed by Fermat of representing nth powers of integers as the sum of two smaller nth powers for n > 2, which was recently solved by Wiles [10].