a/sinA=b/sinb=c/sinc=2R
a+b+c=2R(sinA+sinB+sinc)
max(a+b+c)=2R*3*(sqrt(3)/2)=3*sqrt(3)*R.

方法1：sinx 在(0,pi)是凸函数 f（X1）+f（X2）+f（X3）<=3f[（X1+X2+X3）/3] 即sinA+sinB+sinc<=3sin((A+B+C)/3)=3sqrt(3)/2

方法2：sinA+sinB+sinC =2sin[(A+B)/2]cos[(A-B)/2]+sinC =2cosC/2cos[(A-B)/2]+sinC <=2cosC/2+sinC =2cosC/2(1+sinC/2)
=2sqrt([(1+sinC/2)^3*(1-sinC/2) ) =2sqrt(3)/3 * sqrt((1+sinC/2)(1+sinC/2)(1+sinC/2)*3(1-sinC/2)) 由不等式性质当1+sinC/2=3(1-sinC/2)时，取最大值 即sinC/2=1/2,C=60时，取最大值 全部不等号取等号条件为A=B=C=60 所以最大值为3√3/2.