Geometric explanation of Hutton's formula?

Question

$$\frac{\pi}{4} = 2 \tan^{-1} \frac{1}{3} + \tan^{-1} \frac{1}{7} \;.$$ 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?

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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.
         

Answer 1

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.

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          _{(Image added by J.O'Rourke)}

Answer 2

A similar argument was found by pappus on the French forum les-mathematiques.net.