設三角形ABC邊長為4根號3的正三角形，P為內部一點，若P至三邊的距離分別為a，b，c，試求: 2. a^2 b^3 c^1=?
回答 (1)
2016-02-24 3:12 pm
△ABC = 4√3 (a+b+c)/2
(1/2)sin60°(4√3)² = 4√3 (a+b+c)/2
√3/2 (4√3) = a+b+c
6 = a+b+c = a/2 + a/2 + b/3 + b/3 + b/3 + c ≥ 6 ( (a/2)² (b/3)³ c¹ )^(1/6)
1 ≥ (a/2)² (b/3)³ c¹
108 ≥ a² b³ c¹ , 當且僅當 a/2 = b/3 = c 即 a = 2 , b = 3 , c = 1 時等號成立。
(1/2)sin60°(4√3)² = 4√3 (a+b+c)/2
√3/2 (4√3) = a+b+c
6 = a+b+c = a/2 + a/2 + b/3 + b/3 + b/3 + c ≥ 6 ( (a/2)² (b/3)³ c¹ )^(1/6)
1 ≥ (a/2)² (b/3)³ c¹
108 ≥ a² b³ c¹ , 當且僅當 a/2 = b/3 = c 即 a = 2 , b = 3 , c = 1 時等號成立。
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