Geometric Optimization

Geometry Level 3

If a a , b b , and c c are the side lengths of a triangle opposite to angles A A , B B , and C C respectively, and that a b c = 1 abc =1 , find the minimum value of

1 a 2 sin ⁡ B + 1 b 2 sin ⁡ C + 1 c 2 sin ⁡ A \frac{1}{a^2 \sin B}+\frac{1}{b^2 \sin C} +\frac{1}{c^2 \sin A}

3 3 3\sqrt3 2 3 2\sqrt3 3 2 \frac{3}{2} 3 2 \frac{\sqrt3}{2}

Mohd. Hamza by Mohd. Hamza , 18, India

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1 solution

Chew-Seong Cheong
Feb 15, 2020

Since the three reciprocal terms are positive, we can apply AM-GM inequality as follows:

1 a 2 sin ⁡ B + 1 b 2 sin ⁡ C + 1 c 2 sin ⁡ A ≥ 3 a b c sin ⁡ A sin ⁡ B sin ⁡ C 3 = 2 3 \frac 1{a^2\sin B} + \frac 1{b^2\sin C} + \frac 1{c^2\sin A} \ge 3 \sqrt[3]{\frac {abc}{\sin A\sin B \sin C}} = \boxed{2\sqrt 3}

Equality occurs when a = b = c = 1 a=b=c=1 and A = B = C = 6 0 ∘ A=B=C = 60^\circ .

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