[HN Gopher] An equation that takes Pythagoras to a new level ___________________________________________________________________ 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 ___________________________________________________________________ (page generated 2020-03-14 23:00 UTC)