September 17th, 2010, 06:48 PM  #1 
Senior Member Joined: Apr 2007 Posts: 2,140 Thanks: 0  Pythagorean theorem
Prove that Pythagorean theorem works. Well I know that I need to have three squares on each side of the triangle. 
September 17th, 2010, 07:19 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,805 Thanks: 2150 
The triangle should have a right angle, and have just one square on each side! I think you already know how to prove the theorem.

September 17th, 2010, 11:10 PM  #3 
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: Pythagorean theorem
"Cut the knot." There are infinite algebraic and geometric proofs. 
September 18th, 2010, 12:47 PM  #4 
Global Moderator Joined: May 2007 Posts: 6,784 Thanks: 707  Re: Pythagorean theorem
Trivial corollary  Place any geometric figure on the hypotenuse, then place geometrically similar figures on each leg. The sum of the areas of the figures on the legs equals the area of the figure on the hypotenuse.

September 18th, 2010, 01:16 PM  #5 
Global Moderator Joined: Dec 2006 Posts: 20,805 Thanks: 2150 
That can be achieved by constructing just one line segment, a perpendicular drawn from the rightangle vertex to the hypotenuse, since the resulting smaller triangles have the same angles as the original triangle. Their areas obviously sum to the area of the original triangle, so the Pythagorean theorem follows immediately from the theorem that the areas of similar figures are proportional to the squares on corresponding sides.

September 18th, 2010, 09:16 PM  #6  
Senior Member Joined: Apr 2007 Posts: 2,140 Thanks: 0  Re: Pythagorean theorem Quote:
 
September 19th, 2010, 10:10 AM  #7 
Global Moderator Joined: Dec 2006 Posts: 20,805 Thanks: 2150 
How did you obtain the result for the semicircles without already knowing the Pythagorean theorem?

September 19th, 2010, 06:50 PM  #8 
Senior Member Joined: Apr 2007 Posts: 2,140 Thanks: 0  Re: Pythagorean theorem 
September 20th, 2010, 07:13 AM  #9  
Math Team Joined: Dec 2006 From: Lexington, MA Posts: 3,267 Thanks: 408  Re: Pythagorean theorem Hello, johnny! Quote:
This is my favorite proof . . . Code: a b ***  * *  a  * c *  b  * c *  * * * c *  * *  b a  * c *   * *  *** b a . [color=beige]. . [/color] [color=beige]. . [/color]  
September 20th, 2010, 03:32 PM  #10 
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: Pythagorean theorem
Hello soroban! That is my favorite as well. It is the one I have used over the years that I have had the best success with easily demonstrating the truth of this theorem to students of algebra/geometry. 

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