利用正弦定理化简已知等式得:
(a+c)/b=(a−b)/(a−c),
化简得a^2+b^2-ab=c^2,
即a^2+b^2-c^2=ab,
∴cosC=(a^2+b^2−c^2)/2ab=1/2,
∵C为三角形的内角,
∴C=π/3
(a+b)/c
=(sinA+sinB)/sinC
=2/√3[sinA+sin(2π/3-A)]
=2sin(A+π/6),
∵A∈(0,2π/3),
∴A+π/6∈(π/6,5π/6),
∴sin(A+π/6)∈(1/2,1],
则(a+b)/c的取值范围是(1,2].