Inequalities :::::::::::------

[匿名]

2012-08-09 10:51 pm

Let a,b and c denote the lengths of the three sides opposite to the three angles
A,B and C of a triangle.

(a) Show that
a^2 +b^2 >= 2ab[(√3)/2 sinC +1/2 cosC]

(b) By expressing cosC in terms of a,b and c, show that
a^2+b^2+c^2 >= (4√3)S
where S is the area of the triangle, and that quality holds if and only if triangle ABC
is equilateral.

回答 (1)

用戶2053

2012-08-10 1:59 am

✔ 最佳答案

a) a² + b²
≥ 2ab .................... A.M. ≥ G.M.
≥ 2ab sin(C+30°) ....... Tips : sin(C+30°) ≤ 1
= 2ab ( cos30° sinC + sin30° cosC )
= 2ab (√3/2 sinC + 1/2 cosC)
b)a² + b² ≥ 2ab (√3/2 sinC + 1/2 cosC)
a² + b² ≥ 2ab ( √3/2 sinC + 1/2 (a² + b² - c²)/(2ab) )
4a² + 4b² ≥ 4√3 ab sinC + 2(a² + b² - c²)
4a² + 4b² - 2(a² + b² - c²) ≥ (4√3) 2S ...... Tips : S = (1/2) ab sinC
a² + b² + c² ≥ (4√3)S

From part a) ,
the quality holds if and only if sin(C+30°) = 1 and a² + b² = 2ab.
i.e. C = 60° and a = b.
Hence the quality holds if and only if △ABC is equilateral.

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