A Geometrical Application of Number Theory

Any quadruple of natural numbers {a, b, c, d} is called a "Pythagorean quadruple" if it satisfies the relationship "a[superscript 2] + b[superscript 2] + c[superscript 2]". This "Pythagorean quadruple" can always be identified with a rectangular box of dimensions "a greater than 0," "b greater than 0" and "c greater than 0" in which "d greater than 0" is identifiable with the length of its diagonal. The circumscribing sphere of this rectangular box has an integral diameter length "d greater than 0" corresponding to the "Pythagorean quadruple" {a, b, c, d}. This result extends the well-known "inscribed circle theorem" for any "Pythagorean triple" {a, b, c} of natural numbers "a, b, and c" satisfying "a[superscript 2] + b[superscript 2] = c[superscript 2]." 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 "c greater than 0," and leg lengths "a greater than 0" and "b greater than 0" corresponding to any "Pythagorean triple of natural numbers."