A Personal Pythagorean Proof

          I produced the following proof for the most famous Pythagorean Theorem (perhaps even developed by Pythagoras himself) with a little inspiration from a diagram.  The theorem states that "The square on the hypotenuse of a right triangle is equal to the sum of the squares on the two legs."  The method I used was inspired by a diagram from the Howard Eves text "An Introduction To The History Of Mathematics, Sixth Edition," which demonstrates a possible method used by the Egyptians to prove that the Pythagorean Triple {3, 4, 5}, constitutes the sides of a right triangle.  The diagram (which is actually taken from ancient Chinese geometry) is reproduced, with slight alterations, below.  It shows four congruent right triangles arranged so that the hypotenuse, c, of each form a quadrilateral and the legs, a and b, are placed so that each leg b is applied to the leg a of another congruent triangle and vice versa, so that a smaller inner quadrilateral is formed.

Fig. 10 Area Proof of Pythagorean Theorem

The first step in this proof is to establish that the outer and inner perimeters form squares.
          For the inner quadrilateral,

                    each ∠ lies on a straight ∠ and is supplementary
                          with the opposite right ∠.
                    ∴ ∠ + 90 = 180
                                ∠ = 90
                    As each △ is congruent each side is equal to a - b
                    ∴ Inner perimeter is a square.
          For the outer perimeter,
                    as the △s were taken to be congruent,
                    the ∠s at the vertices are equal to the non-right ∠s of each △
                    ∴ ∠s are complementary (∠ sum △)
                    Each side is equal to c
                    ∴ Outer perimeter is a square
          Now,
                    Area of Large Square = 4 x Area of Triangle + Area of Small Square
                                                   c² = 4ab / 2 + (a - b)²
                                                        = 2ab + a² - 2ab + b²
                                                        = a² + b<sup>2


          EDIT: After further reading, I have discovered that this proof actually appears later in the text from which I drew inspiration under the section on Hindu mathematics.  This disappoints slightly, however I will continue to search for a unique proof (27/11/11).</sup>