In triangle ABC, the condition 1/(1 + sin(A/2)) + 1/(1 + sin(B/2)) + 1/(1 + sin(C/2)) = 2 holds and the side b = 3. The circumradius R of the triangle equals

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sqrt(3). Setting A = B = C = 60 deg gives sin(30 deg) = 1/2, so each term equals 1/(1 + 1/2) = 2/3, and the sum is 3 * (2/3) = 2. This satisfies the constraint. For any degenerate triangle (one angle approaching 0), the constraint cannot be satisfied (as shown by algebraic analysis using x + y + z = pi/2 and Lagrange multipliers, the only real solution inside the triangle is the equilateral one). Hence the triangle must be equilateral with all sides equal to b = 3. By the sine rule, R = b / (2 sin B) = 3 / (2 * sin 60 deg) = 3 / sqrt(3) = sqrt(3).

In triangle ABC, the condition 1/(1 + sin(A/2)) + 1/(1 + sin(B/2)) + 1/(1 + sin(C/2)) = 2 holds and the side b = 3. The circumradius R of the triangle equals

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