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July 23rd, 2011, 12:24 AM  #1 
Member Joined: Jul 2011 From: Europe Posts: 59 Thanks: 2  How would You get to Pythagorean theorem
Hello, my question is probably more philosophical, but I'd like to hear what do you think. Well, from time to time, I like to think about if there is a 100% correct way to think about some subject, in the way that no hidden sense can ran away from you. For example, I expect that we all know Pythagorean theorem, and we also know that it's proven correct. Many of us also understand the nature of proof. But, if it didn't exist (Pythagoras would not discover/publish it), how would we get to it? It may sound funny, but I don't think (for example) I 'd got it that quickly. What do you think? 
July 23rd, 2011, 12:33 AM  #2 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs  Re: How would You get to Pythagorean theorem
The easiest way I know of is to place 4 right triangles together (whose legs are a and b and hypotenuse is c) such that the hypotenuses of the triangles form an interior square, so that we have: 
July 23rd, 2011, 12:42 AM  #3 
Member Joined: Jul 2011 From: Europe Posts: 59 Thanks: 2  Re: How would You get to Pythagorean theorem
Thanks for Your reply, (honestly I didn't know about that, but I just became high school student so I hope it's ok ) well, from the thing you posted could prove it. But I rather meant "how would you think" to get to that idea, that c^2 actually is a^2+b^2 in right trinagle. 
July 23rd, 2011, 12:45 AM  #4 
Member Joined: Jul 2011 From: Europe Posts: 59 Thanks: 2  Re: How would You get to Pythagorean theorem
sorry for typos, I seem not to be able to edit my posts. "well, from the thing you posted could prove it." = "well, the thing you posted could prove it" 
July 23rd, 2011, 12:54 AM  #5 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs  Re: How would You get to Pythagorean theorem
Once you have ten or more posts you will be able to edit your posts...for up to 48 hours. The method I outlined above is what I use to easily show those who know only areas of squares and right triangles how the Pythagorean theorem can be proven. 
July 23rd, 2011, 01:00 AM  #6 
Member Joined: Jul 2011 From: Europe Posts: 59 Thanks: 2  Re: How would You get to Pythagorean theorem
Alright I guess I'll have those in while. "The method I outlined above is what I use to easily show those who know only areas of squares and right triangles how the Pythagorean theorem can be proven. "  correct, but I was asking "how would you get the idea that it really is like that" more, not "how would you prove it when you have the idea". English is not my native language, so I hope it's clear enough. 
July 23rd, 2011, 01:07 AM  #7 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs  Re: How would You get to Pythagorean theorem
Well, that is hard to say, as I knew the result before I ever tried to prove it. All I can say is when presented with a geometric problem, one tries things with the given shape, involving areas, similarity, etc., to arrive at a result. 
July 23rd, 2011, 01:12 AM  #8 
Member Joined: Jul 2011 From: Europe Posts: 59 Thanks: 2  Re: How would You get to Pythagorean theorem
Well, I hope I will be able to complete my problems in that way. I have already tried with sequence of 1+2+3+4+n numbers, but it's hard to make it to correct result when you already know what to do ((n^2+n )/2) 
July 29th, 2011, 03:56 AM  #9 
Member Joined: Jul 2011 From: Europe Posts: 59 Thanks: 2  Re: How would You get to Pythagorean theorem
Hello philipsteele, I am sure they would, but as I stated before, this was rather philosophical question than asking for raw way to get Pythagorean theorem. 
July 29th, 2011, 05:51 AM  #10 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: How would You get to Pythagorean theorem
It's one of the mostproven theorems, next to Quadratic Reciprocity. You can pretty much pick your method. 

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