A,B,C成等差数列,所以B=60°
由正弦定理a/sinA=b/sinB=c/sinC,所以b/sinB=(a+b+c)/(sinA+sinB+sinC)
b=(a+b+c)sinB/(sinA+sinB+sinC)=12sin60°/(sinA+sin60°+sinC)=6√3/(sinA+sinC+√3/2)
sinA+sinC=2sin(A+C)/2cos(A-C)/2=2sin60°cos(A-C)/2=√3cos(A-C)/2
所以当A=C时,cos(A-C)/2取得最大值1,sinA+sinC取得最大值√3.
从而b取得最小值为4.