The linked paper https://fermatslibrary.com/s/proof-without-words-pythagorean-runs https://fermatslibrary.com/s/proof-without-words-pythagorean... gives you the

The linked paper https://fermatslibrary.com/s/proof-without-words-pythagorean-runs 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/EWD975.html 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.