Solution 1
Solving for $R$ yields $R = \tfrac{a\sqrt{bc}}{b+c}$ . We can substitute $R$ into the area formula $A = \tfrac{abc}{4R}$ to get $$\begin{align*} A &= \frac{abc}{4 \cdot \tfrac{a\sqrt{bc}}{b+c} } \\ &= \frac{abc}{4a\sqrt{bc}} \cdot (b+c) \\ &= \frac{(b+c)\sqrt{bc}}{4}. \end{align*}$$ We also know that $A = \tfrac{1}{2}bc \sin(\theta)$ , where $\theta$ is the angle between sides $b$ and $c.$ Substituting this yields $$\begin{align*} \tfrac{1}{2}bc \sin(\theta) &= \frac{(b+c)\sqrt{bc}}{4} \\ 2\sqrt{bc} \cdot \sin(\theta) &= b+c \\ \sin(\theta) &= \frac{b+c}{2\sqrt{bc}} \end{align*}$$ Since $\theta$ is inside a triangle, $0 < \sin{\theta} \le 1$ . Substitution yields $$0 < \frac{b+c}{2\sqrt{bc}} \le 1.$$ Note that $2\sqrt{bc}$ , so multiplying both sides by that value would not change the inequality sign. This means $$0 < b+c \le 2\sqrt{bc}.$$ However, by the AM-GM Inequality , $b+c \ge 2\sqrt{bc}$ . Thus, the equality case must hold, so $b = c$ where $b, c > 0$ . When plugging $b = c$ , the inequality holds, so the value $b=c$ truly satisfies all conditions.
That means $\sin(\theta) = \frac{2b}{2\sqrt{b^2}} = 1,$ so $\theta = 90^\circ.$ That means the only truangle that satisfies all the conditions is a 45-45-90 triangle where $a$ is the longest side. In other words, $(a,b,c) \rightarrow \boxed{(n\sqrt{2},n,n)}$ for all positive $n.$