Can anyone tell me how does Pythagoras discovered his famous theorem - The Pythagoras Theorem?
Does he discover it by drawing the triangle and measure it? Or just by guessing? I want to know because I haven't seen a proof for the theorem by Pythagoras himself; instead I have seen other proofs by other mathematician.

- Pythagoras didn't exist.
- His theroem was known before the ancient greeks.

Wow, I've never heard anyone claim Pythagoras didn't exist before. The way I understood it (but perhaps this is only one of many possibilities) the theorem was actually discovered/proven not by Pythagoras but by one of the Pythagoreans (those are the students of Pythagoras). In fact it is reputed that the discovery partly caused the collapse of his school. Because Pythagoras had built his model of the world in which all quantities are the ratio of two integers (so are called rational in today's terminology). Pythagoras theorem shows that the hypotenuse of the isoceles triangle with sides 1 and 1 is sqrt(2) - and no rational number squares to 2. Therefore rational numbers are insufficient to describe geometry, and so the Pythagoras world model in which rational numbers described everything fell apart.

I don't know how the theorem was first proved (by whoever it was). It is possible that the result was obtained empirically as you suggest but my guess would be that it was discovered at exactly the same time it was proven.

I've heard the theory that Pythagoras didn't exist. Though I've no idea of any of the evidence for or against his existence, so I wouldn't like to comment on whether this is or isn't the case.

All I know is what I have learnt from Dr Bursill-Hall 's excellent History of Mathematics lectures.

He claims that Pythagoras never existed, or that there is practically no evidence for his existence. He also provided much evidence that people knew about the theorem long before the ancient greeks. I don't have his lecture notes with me but I recall in particular an Arabian mud tablet with the expansion of root 2 to a very high order of accuracy.

Very interesting. I'll see if I can find out more.

I think that not much people think about this question before, because we just learn from the school's text book and we just memorize this theorem. It's strange that there is no proof for the Pythagoras theorem in the text book. Even the teachers don't know the proof. So I look into the internet and I found the proof, but not a single one by Pythagoras or the Pythagoreans. Here is the web page's address, hope it helps.
http://www.geocities.com/CapeCanaveral/Launchpad/3740/diagrams.html

You're right - I have only known the proof of Pythagoras for three years whereas I must have known the theorem for a lot longer - I covered it in Year 8 as I remember. But I didn't really understand the concept of a proof before a few years back anyway.

In my view the most attractive geometric proof follows from inscribing a skewed square inside a bounding square and calculating the area.
[This can be found here . - The Editor]

Well I didn't know a proof at all until I followed that link above - but now I do :-)

Perhaps the reason that you can't find the original proof is that no-one knows?

I had thought that Pythagoras proved his theorem by the use of areas in a square. If no one beats me to it, I'll post an image which should give you a good idea. But, then again, this was probably done by another mathematician..

Michael, you mentioned the Pythagoreans, if I recall right, there was also something called the Brotherhood? What was that all about?

One of the various proofs uses the concept of similar triangles. It goes as follows: Assumed that we have a right triangle ABC, right-angled at A construct an altitude AD to BC, triangles ABC and DBA are found to be similar, from which we get AB/BC = AD/AB , OR AB² = AD × BC, etc., etc., etc. - but I have no idea whether this is the proof by Pythagoras (if there was no Pythagoras then by the Mesopotamians who discovered the theorem), or if this is a recent proof. All the same, wonders never cease to be...!!!

There is a biography of Pythagoras on the St Andrew's History of Maths website .
There are many more proofs of the theorem, some of which can be found in Proofs without words - exercises in visual thinking by Roger B. Nelsen, and its sequel, Proofs without words II .