Prove that the calculation yields
$$\begin{aligned} s^{2} & -r^{2}-4 R r \\ & =s^{2}-\frac{\Delta^{2}}{s^{2}}-\frac{a b c}{s} \\ & =s^{2}-\frac{(s-a)(s-b)(s-c)}{s}-\frac{a b c}{s} \\ & =\frac{s^{3}-s^{3}+s^{2}(a+b+c)-s(a b+b c+c a)+a b c-a b c}{s} \\ & =s(a+b+c)-(a b+b c+c a) \\ & =\frac{1}{2}\left(a^{2}+b^{2}+c^{2}\right) . \end{aligned}$$

From $0 < a^{2} + b^{2} + c^{2} \leq 9 R^{2}$, we get
$$0 < s^{2} - (r^{2} + 4 R r) \leq \frac{9 R^{2}}{2},$$

Thus, we have
$$\begin{array}{l} r^{2} + 4 R r < s^{2} \leq r^{2} + 4 R r + \frac{9 R^{2}}{2}, \\ \sqrt{r^{2} + 4 R r} < s \leq \sqrt{r^{2} + 4 R r + 4 R^{2} + \frac{R^{2}}{2}}, \\ 2 \sqrt{r^{2} + 4 R r} < 2 s \leq \sqrt{4(r + 2 R)^{2} + 2 R^{2}} . \end{array}$$