Đáp án:
\[AM = \sqrt {\frac{{{b^2} + {c^2}}}{2} - \frac{{{a^2}}}{4}} \]
Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
\overrightarrow {AM} = \frac{1}{2}\left( {\overrightarrow {AB} + \overrightarrow {AC} } \right)\\
\Leftrightarrow A{M^2} = \frac{1}{4}.\left( {A{B^2} + A{C^2} + 2.\overrightarrow {AB} .\overrightarrow {AC} } \right)\\
\Leftrightarrow A{M^2} = \frac{1}{4}.\left( {A{B^2} + A{C^2} + 2.\cos A.AB.AC} \right)\\
\Leftrightarrow A{M^2} = \frac{1}{4}.\left( {A{B^2} + A{C^2} + 2.\frac{{A{B^2} + A{C^2} - B{C^2}}}{{2.AB.AC}}.AB.AC} \right)\\
\Leftrightarrow A{M^2} = \frac{1}{4}.\left( {2A{B^2} + 2A{C^2} - B{C^2}} \right)\\
\Leftrightarrow A{M^2} = \frac{{A{B^2} + A{C^2}}}{2} - \frac{{B{C^2}}}{4}\\
\Rightarrow AM = \sqrt {\frac{{{b^2} + {c^2}}}{2} - \frac{{{a^2}}}{4}}
\end{array}\)
