# Soft question: For a corner of a circle within a square, why is its ratio equal to $\sqrt{2}$?

Diagrams

^Diagram 1 ^Diagram 2
^Diagram 3

The problem
I was interested in finding the relationship of $JA$ and $CJ$ to the radius of the circle ($JE$). This is because I wanted to test if a circle of a certain radius could fit around the middle square in diagram 3.

To work out $CJ$, this is what I did: $$CJ = CE - JE$$ $$CE = \sqrt{EI^2+FE^2}$$ Since $EI = FE = JE$, $$CE = JE\sqrt{2}$$ $$\therefore CJ = JE\sqrt{2} - JE$$
And to work out $AJ$: $$AJ = JE - AE$$ $$AE = \frac{CE}{2}$$ $$\Rightarrow AE = \frac{JE\sqrt{2}}{2}$$ $$\therefore AJ = JE - \frac{JE\sqrt{2}}{2}$$

And since JE is the radius (r): $$CJ = r\sqrt{2} - r$$ $$AJ = r - \frac{r\sqrt{2}}{2}$$ The weird part
So I got the answers I wanted. However, dividing $CJ$ by $AJ$ gave a result of $\sqrt{2}$. I did do a google search of "corner of circle $\sqrt(2)$" and found that $\sqrt{2}$ seemed very common for circles inscribed within squares. However, in all of the cases, $\sqrt{2}$ would represent some length, not a ratio. I found the results above weird, because it seems that $CJ$ will always be $\sqrt{2}$ times larger than $AJ$. I would have expected $\sqrt{2}$ to have represented some diagonal length, not a ratio/relationship like above. Is there some intuitive or visual explanation for why this is occurring?

$JA$ and $JA'$ are radii of the same circle. Assume $JA=1$, then clearly $CJ=\sqrt{2}$.