Cho  $\Delta ABC$  có các cạnh là  $a, b, c$ và các đường trung tuyến xuất phát từ  $B, C$  là  $m_b, m_c$  thỏa  $\frac{c}{b} =\frac{m_b}{m_c} \neq 1. $
a) Chứng minh rằng:  $2a^2=b^2+c^2$
b) Suy ra rằng:  $2 \cot A=\cot B+\cot C$.
a) Ta có:  $\frac{c}{b}=\frac{m_b}{m_c}      \Leftrightarrow     \frac{c^2}{b^2}=\frac{m_b^2}{m_c^2}     \Leftrightarrow     \frac{c^2}{b^2}=\frac{\frac{1}{2}(a^2+c^2-\frac{b^2}{2} ) }{\frac{1}{2}(a^2+b^2-\frac{c^2}{2} ) }  $
$\Leftrightarrow     \frac{c^2}{b^2}=\frac{2a^2+2c^2-b^2}{2a^2+2b^2-c^2}  $
$\Leftrightarrow     2a^2c^2+2b^2c^2-c^4=2a^2b^2+2b^2c^2-b^4$
$\Leftrightarrow     c^4-b^4+2a^2b^2-2a^2c^2=0$
$\Leftrightarrow     (c^2-b^2)(c^2+b^2-2a^2)=0$
$\Leftrightarrow     c^2+b^2-2a^2=0$   (vì  $c \neq b   \Rightarrow   c^2-b^2 \neq 0$)
$\Leftrightarrow     2a^2=b^2+c^2          (1)$

b) Ta có:  $b^2+c^2=a^2+2bc \cos A$
Do đó   $(1)  \Leftrightarrow    a^2+2bc \cos A=2a^2   \Leftrightarrow     \cos A=\frac{a^2}{2bc} $
Ta lại có:  $\frac{a}{\sin A}=2R    \Rightarrow     a=2R \sin A; b=2R \sin B;  c=2R  \sin C$
Vậy  $\cos A=\frac{a^2}{2bc}=\frac{\sin ^2 A}{2 \sin B \sin C}    \Leftrightarrow     \frac{2 \cos A}{\sin A}=\frac{\sin A}{\sin B \sin C}  $
$\Leftrightarrow     2 \cot A=\frac{\sin (B+C)}{\sin B \sin C}   (A$  bù với  $B+C$)
$\Leftrightarrow     2 \cot A=\frac{\sin B \cos C+\sin C \cos B}{\sin B \sin C} $   
$\Leftrightarrow     2 \cot A=\cot B+\cot C$.
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