Problem

The key insight is that the ratio $\frac{R}{r}$ (circumradius to inradius) is minimized when a triangle is equilateral, and this minimum value is exactly 2. You'll want to express both $R$ and $r$ in terms of the triangle's sides and angles, then use the constraint $\frac{R}{r} \le 2$ to show it forces equality—which only happens for an equilateral triangle. This is a beautiful example of how geometric inequalities can characterize special triangles.

The key insight is that the ratio $\frac{R}{r}$ (circumradius to inradius) is minimized when a triangle is equilateral, and this minimum value is exactly 2. You'll want to express both $R$ and $r$ in terms of the triangle's sides and angles, then use the constraint $\frac{R}{r} \le 2$ to show it forces equality—which only happens for an equilateral triangle. This is a beautiful example of how geometric inequalities can characterize special triangles.