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 Chapter 9 : Right Triangles and Trigonometry Pythagorean Theorem Proofs Pythagorean Theorem Proofs The Pythagorean Theorem, which states that if a triangle is a right triangle, then the sum of the squares of the legs equals the square of the hypotenuse, holds a vital role in mathematics. The theorem is crucial to geometry because it serves as the basis of the definition of distance between two points in a plane, and underlies the proofs of many other mathematical theorems. The first writings about what we now know as the Pythagorean Theorem date back to a Babylonian tablet from the time around 1900-1600 B.C. The tablet shows that the Babylonians had discovered rules for generating Pythagorean triples, groups of three integers for which the square of the larger equals the sum of the squares of the other two. These triples, like 3-4-5, 5-12-13, and 8-15-17, give integer values for the sides of right triangles. The first written proof of the Pythagorean theorem dates from about 300 B.C. to the book the Elements, by Euclid, who taught mathematics in Alexandria, Egypt, an eminent center of learning at the time. Euclid's works are so vast and important that the normal geometry in a plane that we study today is called Euclidean geometry. After stating the Pythagorean Theorem as Proposition 47 in the Elements and then proving it, Euclid stated the converse of the Pythagorean Theorem as Proposition 48 and then proved this statement as well. The converse of a theorem is, in a sense, a statement of the reverse direction of a theorem. For the Pythagorean Theorem, the converse states that if the sum of the squares of the legs of a triangle equals the square of the hypotenuse, then the triangle is a right triangle. Finding new ways of proving the Pythagorean Theorem has remained a fascination over the centuries not only for mathematicians in cultures all around the world but for other imaginative people as well because there are so many widely-varying ways to approach the problem. In some of these approaches, the Pythagorean Theorem is just one special case in a more general theorem. All sorts of diagrams, starting points, and calculations are used in the proofs. One of the hundreds of proofs now in existence is even credited to the great Italian artist Leonardo da Vinci, who possessed great analytical and engineering skills that complemented his incredible artistic abilities. You can find out more about the Pythagorean Theorem and its proofs through The Math Forum a resource for both teachers and students.