acos^2(C/2)+ccos^2(A/2)=3/2b
1/2a(2cos^2(C/2)-1)+1/2a+1/2c(2cos^2(A/2)-1)+1/2c=3/2b
1/2acosC+1/2ccosA+1/2a+1/2c=3/2b
运用正弦定理
可得1/2sinAcosC+1/2sinCcosA+1/2sinA+1/2sinC=3/2sinB
1/2sin(A+C)+1/2(sinA+sinC)=3/2sinB
sinB+sinA+sinC=3sinB
sinA+sinC=2sinB
即a+c=2b
a,b,c成等差数列