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.

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

Scroogey

I need to spend some time myself analyzing your answers, but for $$\theta$$, the [inscribed angle theorem](https://en.wikipedia.org/wiki/Inscribed_angle#:~:text=The%20inscribed%20angle%20theorem%20states,different%20positions%20on%20the%20circle.) should help.

south_korea_ln

science

[Daily puzzle] Find the ratio

I didn't read it that way, but I was wondering if the fact that the triangle is right-isosceles imposes some kind of additional condition that lets us find $$\theta$$.

Don't really have the time to work it further right now though.  Geometry isn't really my forte.