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. So c² = (a−b)²+4A, which on simplifying gives c² = a²+b².

Third proof: These two proofs are in a sense dual to each other, so here is a “self-dual” proof, with the added advantage that it is not necessary to know the value of A. The area of the tilted triangle is the average of the areas of the inner and outer squares, since the difference in either direction is four triangles. So c² = ½[(a+b)²+(a−b)²], which on simplifying gives c² = a²+b². These simple proofs of a not-so-simple theorem call for a couple of philosophical remarks.

• Why three proofs, when one is sufficient to establish the result? Not all of us are perfect logicians; and for the rest of us, three proofs are more convincing than one. “What I tell you three times is true”, as the Bellman said.
• As with any proof using a diagram, certain things are obvious from the diagram but a formal proof lies rather deeper. For example, why do congruent triangles have the same area? This depends on the fact that Euclidean transformations (translations, rotations, reflections, etc.) preserve area; so, eventually, it is a question of group theory amd measure theory.