Three easy proofs of Pythagoras’ Theorem

Take a right-angled triangle with hypotenuse c and the other two sides a and b. Pythagoras’ Theorem tells us that c2 = a2+b2. Let the area of the triangle be A. We know that A = ab/2 (since an a×b rectangle is cut into two such triangles by a diagonal). Here are three simple proofs, not in the least original. All use the same diagram, constructed as follows. Take a square with side a+b. At each corner, measure a distance a along the side in the clockwise direction and b along the side in the anticlockwise direction. Complete these two segments to a rectangle, and draw the diagonal not containing the corner where you began. You will see, in addition to the original square, a tilted square with side c, and an inner small square with side a−b. First proof: The outer square is made up of the tilted square and four copies of the triangle. So (a+b)2 = c2+4A, which on simplifying gives c2 = a2+b2. Second proof: The tilted sqiare is made up of the inner sauqre and four copies of the triangle.…