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.

south_korea_ln

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 

science

[Daily puzzle] Find the ratio

Ah, yes!

![angles](https://i.ibb.co/Z8WcPLq/c.png)

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, $$\frac{r}{k} = \sqrt{2-\sqrt{2}} \approx{0.76537}$$