Squares and Circles

If $r$ is the radius of the blue circle, $R$ the radius of the large quarter circles, $s$ the radius of the red circle, and $X$ the side of the small square (the big square has side $R$ ), then we obtain the following equations $\begin{aligned} (R+r)^2 & = \; (R-r)^2 + \tfrac14R^2 \\ R^2 & = \; X^2 + \tfrac14(X+R)^2 \\ (R-s)^2 & = \; (X+s)^2 + \tfrac14R^2 \end{aligned}$ which solve to give $R = 16r$ , $X = \tfrac35R$ and $s = \tfrac{39}{320}R$ , so that $s = \tfrac{39}{20}r$ , making the answer $\boxed{20\,:\,39}$ .