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Correct derivation: The height $ h $ of an equilateral triangle is $ rac{\sqrt{3}}{2}s $. The centroid (which coincides with the circumcenter) is at $ rac{2}{3} $ of the height from a vertex, so:
R = rac{2}{3} \cdot rac{\sqrt{3}}{2}s = rac{s}{\sqrt{3}} \cdot rac{\sqrt{3}}{\sqrt{3}} = rac{s\sqrt{3}}{3}
R = rac{s\sqrt{3}}{3}

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📰 Solution: To minimize 📰 under the constraint $ x + y + z = 1 $, we apply the **Cauchy-Schwarz Inequality** in the following form: 📰 Equality occurs when $ \frac{x}{y} = \frac{y}{z} = \frac{z}{x} $, which implies $ x = y = z $. Since $ x + y + z = 1 $, we have $ x = y = z = \frac{1}{3} $.