4 ms·

This is the most succcinct proof, it's an ancient Chinese visual proof: https://www.dbai.tuwien.ac.at/proj/pf2html/proofs/pythagoras/pythagoras/pythagoras7.gif

This is the most succcinct proof, it's an ancient Chinese visual proof: https://www.dbai.tuwien.ac.at/proj/pf2html/proofs/pythagoras/pythagoras/pythagoras7.gif https://www.dbai.tuwien.ac.at/proj/pf2html/proofs/pythagoras... It's not labelled this way but notice that the BIG squares are identical, having sides of a+b. Only the four triangles are rearranged. Just subtract the four identical triangles. In the second arrangement, a c^2 area is present in addition to the same four triangles. In the first arrangement this had been rearranged to an a^2 and a b^2 area. It's important to assure yourself the triangles are the same and the big square is the same - there is no tiny hidden sliver or something, and right angles are preserved, there are no shenanigans. second explanation of same: http://www.math.toronto.edu/colliand/notes/pythagoras.html http://www.math.toronto.edu/colliand/notes/pythagoras.html

ndr 9y ago
This is beautiful. To be honest the second image with the tilted c² is enough. From that picture alone you can figure the outer square has the same area as the inner square + 4 times the triangle area: (a+b)² = c² + 4(ab/2) a² + b² + 2ab = c² + 2ab a² + b² = c²
- logicallee 9y ago
  Thanks. Your proof works, I could follow it. Yours does use some elementary algebra[1], which wasn't used in the Chinese geometric proof. I wonder if ancient Chinese mathematicians could simplify (a+b)^2 symbolically, or even did symbolic algebra this way? (When I referred to "ancient Chinese visual proof" I wasn't bullshitting, but I didn't find a source with the exact 2-part picture, though this makes same claim using same pictures: http://www.researchhistory.org/2012/10/24/earliest-evidence-of-pythagoras-theorem/ http://www.researchhistory.org/2012/10/24/earliest-evidence-... ) I'm sure they knew that the area of triangles, which you also use, is (ab/2) - but the purely visual proof needs nothing more than the knowledge that the area of a square is the square of the length of its sides. (And I guess some obvious facts like that no matter how you divide an area the sum of the areas of its parts will be same - the reason I mention tiny slivers is in some fake geometric proofs this intuitive knowledge is abused. For example, see this excellent description: https://en.m.wikipedia.org/wiki/Missing_square_puzzle https://en.m.wikipedia.org/wiki/Missing_square_puzzle Before you open the solution, you can look at the trick for as long as you want, you won't figure it out.) [1] https://en.m.wikipedia.org/wiki/FOIL_method https://en.m.wikipedia.org/wiki/FOIL_method

ndr 9y ago

This is beautiful. To be honest the second image with the tilted c² is enough. From that picture alone you can figure the outer square has the same area as the inner square + 4 times the triangle area: (a+b)² = c² + 4(ab/2) a² + b² + 2ab = c² + 2ab a² + b² = c²

logicallee 9y ago
Thanks. Your proof works, I could follow it. Yours does use some elementary algebra[1], which wasn't used in the Chinese geometric proof. I wonder if ancient Chinese mathematicians could simplify (a+b)^2 symbolically, or even did symbolic algebra this way? (When I referred to "ancient Chinese visual proof" I wasn't bullshitting, but I didn't find a source with the exact 2-part picture, though this makes same claim using same pictures: http://www.researchhistory.org/2012/10/24/earliest-evidence-of-pythagoras-theorem/ http://www.researchhistory.org/2012/10/24/earliest-evidence-... ) I'm sure they knew that the area of triangles, which you also use, is (ab/2) - but the purely visual proof needs nothing more than the knowledge that the area of a square is the square of the length of its sides. (And I guess some obvious facts like that no matter how you divide an area the sum of the areas of its parts will be same - the reason I mention tiny slivers is in some fake geometric proofs this intuitive knowledge is abused. For example, see this excellent description: https://en.m.wikipedia.org/wiki/Missing_square_puzzle https://en.m.wikipedia.org/wiki/Missing_square_puzzle Before you open the solution, you can look at the trick for as long as you want, you won't figure it out.) [1] https://en.m.wikipedia.org/wiki/FOIL_method https://en.m.wikipedia.org/wiki/FOIL_method