A geometrical application of number theory

Any quadruple of natural numbers is called a Pythagorean quadruple if it satisfies the relationship . This Pythagorean quadruple can always be identified with a rectangular box of dimensions in which is identifiable with the length of its diagonal. The circumscribing sphere of this rectangular box has an integral diameter length corresponding to the Pythagorean quadruple .This result extends the well-known ‘inscribed circle theorem’ for any Pythagorean triple of natural numbers satisfying . This above-mentioned theorem asserts the positive integer nature of the radius of the inscribed circle, that is associated with any right triangle with hypotenuse length , and leg lengths corresponding to any Pythagorean triple of natural numbers.