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Ah, yes!
The two green
c have equal lengths.Hence, the angles in the bottom right corner must be equal.
They both add up to 45°, because the triangle is right-isosceles, hence they are
\frac{45°}{2}.Now the inscribed angle theorem says
\theta = 3*45°.Therefore,
\frac{r}{k} = \sqrt{2-\sqrt{2}} \approx{0.76537}reply
I have
\frac{\sqrt{2+\sqrt{2}}}{\sqrt{2}+1} which also \approx 0.76537 if I'm not mistaken.Your expression is even simpler.
Good job :)
A GitHub PR for collapsable answers is on its way in next few days. We'll be able to hide spoilers which should increase the fun factor.
reply
\theta, the inscribed angle theorem should help.