Exactly Exchemist. And our contradiction arises here: "because √(a²+b²) is NOT (a+b )." The difference is how we read that. I know you are right. And I know why it looks wrong to me. My problem comes with the symbol that encloses the two sides squared. It says, as I think you also said "the square root of". Isn't the square root the base number? So, wouldn't the square root of a-squared be a?

There is a word for what I am doing - arguing against myself - and I apologize for that. You and I put it in the same words but I am wrong in my conclusion and I can't see why. I could go on with the triangle I have drawn to illustrate but it would get boring to you.

Let me think about it for a while. Maybe it will come to me at 2:00 AM. Thank you for the reading. That does confirm what I thought it said.

P.S. Suddenly, I saw this. "If you square (a+b ) you get a² **+ 2ab** + b²." I didn't see that before but it is a "poser" for sure. Thanks, I think. :-)

Just saw your PS. Yes, this is something in algebra that you need to know.

When you multiply two brackets together , like (a + b ) x ( c + d), you have to multiply **each** term in the second by** each** term in the first, so you get **4 terms**, like this: ac + ad from multiplying by the first, PLUS bc + bd, from multiplying by the second, i.e. ac + ad + bc + bd in total.

So, with (a+b )(a+b ) you get: a.a + a.b + b.a + b.b, in other words a² + 2ab + b².

Again, to give a numerical example, let's look at (2+3)² = 5² = 25. Applying the rule above to (2+3)(2+3) you would expand the brackets to get: 2.2 + 2.3 +3.2 + 3.3 = 4 + 6 + 6 + 9 = 25. Bingo!

It is a famous schoolboy error to think (a+b )² = a² + b² . If you try that in the numerical example above you get 2.2 + 3.3, which gives you only 13. Fail!

**Edited by exchemist, 18 June 2017 - 11:37 AM.**