Square size from a 5 by 10 box touching its incircle

A circle sits inside a square, touching all four sides. In one corner of the square a $5\times10$ rectangle is placed with two sides along the square's edges, and its opposite vertex lies exactly on the circle. How long is the square's side?

$50$. Put the corner at the origin, so the circle has centre $(r,r)$ and radius $r$. The vertex $(10,5)$ on the circle gives $(r-10)^2+(r-5)^2=r^2$, i.e. $r^2-30r+125=0$, so $r=25$ (the root $r=5$ would make the rectangle span the whole side). The side is $2r=50$.