TITLE: Geometric explanation of Hutton's formula?
QUESTION [6 upvotes]: $$\frac{\pi}{4} = 2 \tan^{-1} \frac{1}{3} + \tan^{-1} \frac{1}{7} \;.$$
Is there some geometric construction that explains this beautiful equation
(known as Hutton's formula)?
Perhaps a "proof without words" figure that makes it self-evident?

Here is Figure 1a from the reference Henry provided:

Nelsen, Roger B. "Proof Without Words: The Formulas of Hutton and Strassnitzky." Mathematics Magazine 86 5 (2013): 350-350.

REPLY [14 votes]: All of these Machin-like formulas have proofs-without-words. First, notice that there is a one line proof using the Gaussian integers:
$$(3+i)^2 (7+i) = 50 + 50 i.$$
Taking arguments of both sides proves the result (modulo $2 \pi$).
Now, plot the products
$$3 \times 3 \times 7=63,\qquad  (3+i) \times 3 \times 7=63+21 i,$$
$$(3+i)^2 \times 7 = 56+42i,\qquad (3+i)^2 (7+i) = 50+50 i.$$
Here the $3$'s and $7$'s are the real parts of $3+i$ and $7+i$.
Each consecutive pair of complex numbers forms a right triangle with the third vertex at $0$:
$$(0,63, 63+21i),\ (0, 63+21i, 56+42i),\ (0, 56+42i, 50+50 i).$$
Draw each of those triangles and you have a proof without words. 

 
 
 
 
 


 
 
 
 
 
(Image added by J.O'Rourke)