This theorem states that in any right triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides: r2 = x2 + y2. More than 100 different proofs have been given for this extremely important theorem of euclidean plane geometry, which is named for the Greek philosopher and mathematician Pythagoras (sixth century B.C.). One of these is based on a comparison of areas in the squares in Fig. 1.
The three-dimensional Pythagorean theorem (Fig. 2) may be phrased “the square of the diagonal of a rectangular box is equal to the sum of the squares of three adjacent edges that meet at a vertex: r2 = x2 + y2 + z2.”
Squares that are equal to the sum of two or three squares were known to Pythagoras. For example, 32 + 42 = 52 and 12 + 42 + 82 = 92. The name pythagorean triple is given to any triple (x, y, r) of whole numbers such that x2 + y2 = r2, and the name Pythagorean quadruple to such integral quadruples (x, y, z, r) for which x2 + y2 + z2 = r2. All pythagorean triples in which x, y, r are without common factor and x is odd are obtained by replacing the letters a and b in the triple (a2 − b2, 2ab, a2 + b2) by whole numbers that have odd sum and no common factor; and similarly all such pythagorean quadruples are found by replacing a, b, c, d in the quadruple (a2 − b2 + c2 − d2, 2ab + 2cd, 2bc − 2ad, a2 + b2 + c2 + d2) by whole numbers that have odd sum and no common factor.