已知外接圆半径为6的△ABC的三边a,b,c,S=a^2-(b-c)^2

S=a^2-(b-c)^2=a^2-b^2-c^2+2bc
据余弦定理：S=-2bccosA+2bc
又 ：S=0.5bcsinA
4(1-cosA)=sinA
8sin^A/2=2sinA/2cosA/2
sinA/2(cosA/2-4sinA/2)=0
tanA/2=1/4
tanA=2tanA/2/(1-tan^2A/2)
=(1/2)/(1-1/16)=8/15
sinA=(8/15)/√(1+64/225)
=8/17
(2)
S=3(a+b+c)=3a[1+(b+c)/a]
=3a[1+(sinB+sinC)/sinA]
=3a(1+(4*17)/(3*8)]
=23a/2
=a^2-(b-c)^2
=a^2[1-(17^2/8^2)(4/3-2sinC)^2]
a=72*23/[144-289(2-3sinC)^2
sinC=2/3时
amax=(72*23)/(9*16)=23/2
Smax=23a/2=23^2/4=529/4

S=a^2-(b-c)^2=-(b^2+c^2-a^2)+2bc=1/2bcsinA
联立4(1-cosA)=sinA,sin^2A+cos^2A=1,A≠0
解得cosA=15/17,sinA=8/17
b=2R*sinB,c=2R*sinC
b+c=12(sinB+sinC)=16
bc=b(16-b)=-(b-8)^2+64
s=1/2bcsinA=4/17[-(b-8)^2+64]
当b=8，sinB=2/3时
Smax=256/17