Cho tam giác ABC có độ dài các cạnh là a, b, c the...

A

- Hướng dẫn giải

Ta có:

\(\begin{array}{l}a + c = 2b \Leftrightarrow \sin A + \sin C = 2\sin B\\ \Leftrightarrow 2\sin \frac{{A + C}}{2}c{\rm{os}}\frac{{A - C}}{2} = 4\sin \frac{B}{2}.c{\rm{os}}\frac{B}{2} = 4\sin \frac{{A + C}}{2}.c{\rm{os}}\frac{{A + C}}{2}\\ \Leftrightarrow c{\rm{os}}\frac{{A - C}}{2} = 2c{\rm{os}}\frac{{A + C}}{2} \Leftrightarrow c{\rm{os}}\frac{A}{2}c{\rm{os}}\frac{C}{2} + \sin \frac{A}{2}\sin \frac{C}{2} = 2c{\rm{os}}\frac{A}{2}c{\rm{os}}\frac{C}{2} - 2\sin \frac{A}{2}\sin \frac{C}{2}\\ \Leftrightarrow 3\sin \frac{A}{2}\sin \frac{C}{2} = c{\rm{os}}\frac{A}{2}c{\rm{os}}\frac{C}{2} \Leftrightarrow 3\tan \frac{A}{2}\tan \frac{C}{2} = 1 \Leftrightarrow \tan \frac{A}{2}\tan \frac{C}{2} = \frac{1}{3}\end{array}\)