m+n=(cos3A/2+cosA/2,sin3A/2+sinA/2)
|m+n|
=√[(cos3A/2+cosA/2)^2+(sin3A/2+sinA/2)^2]
=√[(cos3A/2)^2+(cosA/2)^2+2cos3A/2cosA/2+(sin3A/2)^2+(sinA/2)^2+2sin3A/2sinA/2]
=√[2+2(cos3A/2cosA/2+sin3A/2sinA/2)]
=√[2+2cos(3A/2-A/2)]
=√(2+2cosA)
=√3
cosA=1/2、A=π/3
b+c=2a
(b+c)^2=b^2+c^2+2ac=4a^2
b^2+c^2-a^2=3a^2-2bc
cosA=(b^2+c^2-a^2)/(2bc)=3a^2/(2bc)-1=1/2
3a^2/(2bc)=3/2
a^2=bc、a=√(bc)
b+c=2√(bc)
(√b-√c)^2=0、b=c
又A=π/3
所以△ABC是等边三角形.