TITLE: Proving :$\arctan(1)+\arctan(2)+ \arctan(3)=\pi$
QUESTION [11 upvotes]: Possible Duplicate:
Why does $\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\pi$? 

How to prove $$\arctan(1)+\arctan(2)+ \arctan(3)=\pi$$

REPLY [72 votes]: Once I have seen a very nice proof of this: your claim is equivalent to proving that the sum of red, green and blue angles is $\pi$. Note that in the second picture, the blue-green triangle is right and isosceles (that is 45-45-90 triangle and thus similar to small red-black triangle in the first diagram).

Cheers!