A Half-Angle Lower Bound

★ 8/10Law of Sines trigonometry Substitution Trigonometric Identities inequality

Problem

The triangle $ABC$ has angles $A, B, C$ with $C \geqslant \dfrac{\pi}{3}$ , and sides $a, b, c$ . Prove that

(a + b)\left(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right) \geqslant 4 + \frac{1}{\sin\dfrac{C}{2}}.

Answer

Solution

Difficulty	8/10
Topics	Law of Sines, trigonometry, Substitution, Trigonometric Identities, inequality

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