9

I recently found a fun proof of $\arctan (1)+\arctan (2)+\arctan (3)=\pi$ from this answer

enter image description here

It's an "Aha" moment for me as I realized the proof's simplicity, still I was surprised by the result summing up to $\pi$.

What are some results that are straightforward to understand but have outcomes that are unexpectedly profound or interesting?

Thomas Andrews
  • 186,215
Danny Wen
  • 322
  • Do you want especialy graphical proofs ? If such is the case, see here – Jean Marie Aug 03 '24 at 19:42
  • Why is arctan(red angle in 2nd picture) $= 1$ true? – Mike Aug 03 '24 at 19:45
  • 2
    Nevermind I see it. The two green segments in 2nd picture have the same length and are perpendicular [the slope of one is $2$ and the slope of the other is $-1/2$ so they must be orthogonal]. Nice picture. – Mike Aug 03 '24 at 19:48
  • $$\operatorname{arccot}(1)+\operatorname{arccot}(2)+\operatorname{arccot}(3)=\fracπ2$$ – Gwen Aug 03 '24 at 23:15
  • I like the proof that the area of a regular dodecagon whose radius is $1$ m is $3$ m$^2$ – ajotatxe Aug 04 '24 at 00:19
  • Numberphile has a YouTube video on this problem/solution: The Three Square Geometry Problem. – awkward Aug 04 '24 at 12:59
  • These are very interesting! The regular dodecagon fact just made my day, sorry mods decided to close it. I think these soft questions has a place here, because without pointing them out, I might never notice them. Just like this one about graphical proofs: https://math.stackexchange.com/questions/733754/visually-stunning-math-concepts-which-are-easy-to-explain – Danny Wen Aug 04 '24 at 20:25

0 Answers0