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.