Pythagorean Proof

An elegant proof of Pythagoras' theorem for trigonometry was created just after the turn of the first millennium by the Indian Mathematician Bhaskara. Two versions of it, with specific examples, are given below. He started with the basic geometry facts that the geometric area of a square is the length of one side squared and the area of a triangle is the product of its base times its height, multiplied by one-half. As a specific example, he inscribed a small square inside a larger 5 x 5 square, so each corner of the inner square coincides with a point on the sides of the outer square. From inspection the outer square's area is seen to contain exactly four equal right triangle, plus the inner square. The hypotenuse of each of the four right triangles is one side of the inner square. If the value of each right triangle's base is 3 and the value of the height is 2, the total of all the four triangle's areas is 4 times 2 times 3 times 1/2 which is equal to 12. The area of the outer square is 5 times 5, which is 25. The inner square's area can be found by subtracting the the area of the outer square from the four triangle's area. So, the inner square's area is equal to 25 minus 12 which is 13. According to the Pythagorean Theorem the hypotenuse squared is equivalent to the sum of the two sides squared, and here 2 squared plus 3 squared also equals 13. This proves the hypotenuse equals the square root of 13 and the addition of the two sides squared is equal to the hypotenuse squared. This specific proof will work with any two inscribed squares and can be expanded into a general proof of he theorem.

Math Mysteries: The right triangle used in the example above is a perfect Pythagorean triple. These are the cases where all three sides are three integers and here all three are in a row, 3² + 4² = 5². The angle for this perfect conjunction is approximately 37 degrees. This is also the first basic triple which is a set of any three integers that satisfy the theorem. All multiples of a basic triple are also triples, such as: 6² + 8² = 10² and 9² + 12² = 15² which are all multiples of the first basic set. The next set of basic triples are the integers 8² + 15² = 17² and the next basic set after that is 12² + 35² = 37². The mystery is why all the basic triples subscribe to the formula 2p, p² +1, p² - 1, where p is an even integer.