Unit circle, midpoint $O(0,0)$, points $P(\cos \alpha,\sin \alpha)$ and $A(\cos \alpha,0)$. Midpoint of the incircle $M(\ (\cos \frac{\alpha}{2}-\sin \frac{\alpha}{2})\cos \frac{\alpha}{2},(\cos \frac{\alpha}{2}-\sin \frac{\alpha}{2})\sin \frac{\alpha}{2})$. Radius of this incircle $r=(\cos \frac{\alpha}{2}-\sin \frac{\alpha}{2})\sin \frac{\alpha}{2}=f(\alpha)$. Maximum if $f'(\alpha)=0$, if $\cos \alpha-\sin \alpha=0$, if $\alpha=\frac{\pi}{4}$. Maximum radius $=\frac{\sqrt{2}-1}{2}$.