Pythagoras's Theorem

Here is another pencil, I said, handing him one. Do continue.

He looked pained. Then he began to concentrate hard. He drew a very faint horizontal line out from the base line of the triangle, beginning at the right angled end. Both he and I knew at once that wasn't right.

It will come back to me, he muttered. It is a long time since I've done it.

How are you doing? I asked sweetly. Would you like a rubber?

No, thank you. Now, if I extend a vertically and b horizontally ....... Yes! I have it! Extend the lines until they are at a point which forms a right angled triangle with a corner of the square! Complete the larger square! Now I have a large square composed of a smaller square and 4 equal sized right angled triangles. That's it.

Go on then, you're only half way there.

He looked at his square. I looked at his square. I looked at him.

You don't remember it do you? I said.

No, he admitted. It temporarily eludes me.

Me too, I said. Let's figure it out together. One thing I do remember; it's logical. And it's to do with working out the area of the larger square in two different ways.

Yes! said Pliny excitedly. The area of the larger square is (a+b) squared. But it is also c squared (which is the area of the smaller square,) plus 4 times half ab, ( half ab being the area of each of the 4 right angled triangles surrounding the smaller square).

Of course! I said. So ( a+b) squared must be equal to 4 times half ab plus c squared. In other words, a squared plus 2ab plus b squared equals 2ab plus c squared.

Take 2 ab from each side! roared Pliny. Which leaves: a squared plus b squared equals c squared! I've done it! The square on the hypotenuse equals the sum of the squares on the other 2 sides! What's the matter? he said, looking at me.

That used to be my party trick, I said dejectedly. At least that was why I learned it.

But, said Pliny, you hardly ever go to parties, nor do you strike me as the sort of person who would offer to perform a party trick of a mathematical nature for the entertainment of others.

I'm not, I replied. And I can never remember it anyway.

Never mind, said Pliny kindly. I am much the same. Here's your pencil, now practice on your own.