2.
a, Theo định lí hàm số sin:
\(\dfrac{a}{sinA}=\dfrac{b}{sinB}=\dfrac{c}{sinC}\Rightarrow sinA=\dfrac{b}{a}sinB=\dfrac{c}{a}sinC\)
Khi đó: \(2sinA=sinB+sinC\)
\(\Leftrightarrow2sinA=\dfrac{b}{a}sinA+\dfrac{c}{a}sinA\)
\(\Leftrightarrow2=\dfrac{b}{a}+\dfrac{c}{a}\)
\(\Leftrightarrow b+c=2a\) đúng theo giả thiết
\(\Rightarrowđpcm\)
b, Ta có \(S_{ABC}=\dfrac{1}{2}a.h_a=\dfrac{1}{2}b.h_b=\dfrac{1}{2}c.h_c\)
\(\Rightarrow\left\{{}\begin{matrix}h_a=\dfrac{2.S_{ABC}}{a}\\h_b=\dfrac{2.S_{ABC}}{b}\\h_c=\dfrac{2.S_{ABC}}{c}\end{matrix}\right.\)
Khi đó: \(\dfrac{2}{h_a}=\dfrac{1}{h_b}+\dfrac{1}{h_c}\)
\(\Leftrightarrow\dfrac{2}{\dfrac{2.S_{ABC}}{a}}=\dfrac{1}{\dfrac{2.S_{ABC}}{b}}+\dfrac{1}{\dfrac{2.S_{ABC}}{c}}\)
\(\Leftrightarrow\dfrac{a}{S_{ABC}}=\dfrac{b}{2.S_{ABC}}+\dfrac{c}{2.S_{ABC}}\)
\(\Leftrightarrow b+c=2a\) đúng theo giả thiết
\(\Rightarrowđpcm\)
