By the Law of Cosines: $$r^2 = 1^2 + (r^2)^2 -2\cdot 1 \cdot r^2\cdot\cos (120^\circ -2\alpha) $$ or \begin{equation} r^4 + 1 = r^2 + 2\cdot r^2\cdot\cos (120^\circ -2 \alpha) \end{equation} Meanwhile, because $r^4 + 1 \ge 2r^2$, therefore, \begin{align*} &r^2 + 2\cdot r^2\cdot\cos (120^\circ - 2\alpha)\ge 2r^2\\ \implies\quad&\cos(120^\circ-2\alpha)\ge\frac{1}{2}\\ \implies \quad&60^\circ > \alpha \ge 30^\circ \end{align*} By the Law of Sines: \begin{align*} \frac{r^2}{1}&=\quad\frac{\sin(60^\circ + \alpha)}{\sin\alpha}\\ &=\quad\frac{\sin 60^\circ \cos\alpha + \cos 60^\circ \sin \alpha}{\sin\alpha}\\ &=\quad\frac{1}{2}+\frac{\sqrt{3}}{2}\cdot\cot\alpha\\ &\le\quad\frac{1}{2}+\frac{\sqrt{3}}{2}\cdot\sqrt{3}\\ &=\quad 2 \end{align*} $$\therefore\quad r \le \sqrt{2}$$