1.
因cos^2(α/2)=(1+cosα)/2
所以
a*cos^2(C/2)+c*cos^2(A/2)=(3/2)b
a*(1+cosC)/2+c*(1+cosA)/2=(3/2)b
acosC+ccosA=3b-a-c
b=3b-a-c
2b=a+c
证毕.
2.
b^2=a^2+c^2-2accosB
(a+c)^2=4a^2+4c^2-8accosB
2ac=3a^2+3c^2-8accosB
8accosB=3a^2+3c^2-2ac
3a^2+3c^2≥6ac
3a^2+3c^2-2ac≥4ac
8accosB≥4ac
cosB≥1/2
B≤π/3