[HN Gopher] An equation that takes Pythagoras to a new level
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       An equation that takes Pythagoras to a new level
        
       Author : soheilpro
       Score  : 37 points
       Date   : 2020-03-14 01:40 UTC (21 hours ago)
        
 (HTM) web link (medium.com)
 (TXT) w3m dump (medium.com)
        
       | chx wrote:
       | The linked paper https://fermatslibrary.com/s/proof-without-
       | words-pythagorean... gives you the formula and I am much more a
       | symbols guy than a geometry guy so I will prove it that way,
       | first rearrange it:
       | 
       | (4T_n)^2=(4T_n+1)^2-(4T_n-1)^2+....(4T_n+n)^2-(4T_n-n)^2
       | 
       | Now, (4T_n+k)^2-(4T_n-k)^2=16T_nk (this, of course, shows up in
       | the geometric proof as four rectangles where one side is 4T_n and
       | the other is k)
       | 
       | so this is equivalent to
       | 
       | 16T_nT_n = 16 T_n (1+...n)
       | 
       | which is equvialent to:
       | 
       | T_n=1+...n
       | 
       | But that is the very definition of T_n. Q.e.d.
       | 
       | Also if you want to take Pythagoras to a new level, Edsger W.
       | Dijkstra who is much better known for his work in CS has a very
       | interesting formulation
       | https://www.cs.utexas.edu/users/EWD/transcriptions/EWD09xx/E...
       | proving sgn(alpha+beta-gamma)=sgn(a^2+b^2-c^2) which includes and
       | extends the Pythagoras theorem.
        
       | jimhefferon wrote:
       | The caption "The equation 102 + 112 + 122 = 132 + 142, whose
       | answer is that both sides equal 365, was immortalized in a
       | different form in this 1895 painting: "Mental Arithmetic. In the
       | Public School of S. Rachinsky." (NIKOLAY BOGDANOV-BELSKY)" has
       | funky exponents. Medium requires that I make an account to say
       | that, so I am saying it here.
        
         | mkl wrote:
         | That formatting error is repeated in at least one other place
         | too.
        
         | jkinudsjknds wrote:
         | On the subject of that painting, can anyone tell me what the
         | kids are wearing on their feet?
        
       | Avshalom wrote:
       | Ptolemy's theorem is another Pythagoras on steroids
       | 
       | https://m.youtube.com/watch?v=bJOuzqu3MUQ
        
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       (page generated 2020-03-14 23:00 UTC)