Pythagoras' Theorem is probably the best-known mathematical theorem. Even most non-mathematicians have some vague idea that it involves triangles and squaring something known as the hypotenuse.

Because the ideas in the previous chapters on ‘How to read a theorem’ and ‘How to read a proof’ are so important we will apply them to this famous theorem to see them in action. So in this chapter we will pull apart the theorem and its proof, we'll see a converse for it and also a generalization.

Statement of Pythagoras' Theorem

As you are a budding mathematician, you probably have a better idea than a non-mathematician of what the statement is, but here it is again.

Theorem 19.1

For a right-angled triangle the square of the hypotenuse is equal to the sum of the squares of the other sides.

Exercise 19.2

Use the ideas from Chapter 16, How to read a theorem, to analyse the theorem. Compare your analysis with the one given below.

Study of the theorem

We now analyse the theorem as though we were meeting it for the first time. Obviously we would check what all the words mean, for example, what is a hypotenuse? This is fairly obvious, but what about the other techniques in Chapter 16? We shall apply them now.