imc

You can express the line connecting the centers of an outer circle and the inner circle in two different ways. You can add the radius of both circles to get $r+1.$ You can also add the radius of two outer circles and use a $45-45-90$ triangle to get $\frac{2r}{\sqrt{2}}=r\sqrt{2}.$ Since both expressions represent the same length, you can set them equal to each other. $$\begin{array}{rl}r+1& =r\sqrt{2}\\ 1& =r(\sqrt{2}-1)\end{array}$$ $$r=\frac{1}{\sqrt{2}-1}=\frac{1(\sqrt{2}+1)}{(\sqrt{2}-1)(\sqrt{2}+1)}=\frac{\sqrt{2}+1}{2-1}=\boxed{1+\sqrt{2}}$$