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(a A+bB+cC)/(a+b+c) >= \pi /3

Source: Spanish Mathematical Olympiad 1971 P4

December 5, 2022
geometryinequalitiesgeometric inequalityalgebratrigonometry

Problem Statement

Prove that in every triangle with sides a,b,ca, b, c and opposite angles A,B,CA, B, C, is fulfilled (measuring the angles in radians) aA+bB+cCa+b+c≥π3\frac{a A+bB+cC}{a+b+c} \ge \frac{\pi}{3}
Hint: Use a≥b≥c⇒A≥B≥Ca \ge b \ge c \Rightarrow A \ge B \ge C.
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