Ta có:
\(\overrightarrow{DB}=\overrightarrow{DA}+\overrightarrow{AB}\)
\(\overrightarrow{AM}=\dfrac{1}{2}\overrightarrow{AB}+\dfrac{1}{2}\overrightarrow{AC}=\dfrac{1}{2}\overrightarrow{AB}+\dfrac{1}{2}\overrightarrow{AD}+\dfrac{1}{2}\overrightarrow{DC}\)
Để \(BD\perp AM\Rightarrow\overrightarrow{DB}.\overrightarrow{AM}=0\)
\(\Rightarrow\left(\overrightarrow{DA}+\overrightarrow{AB}\right)\left(\dfrac{1}{2}\overrightarrow{AB}+\dfrac{1}{2}\overrightarrow{AD}+\dfrac{1}{2}\overrightarrow{DC}\right)=0\)
\(\Leftrightarrow\overrightarrow{DA}.\overrightarrow{AB}+\overrightarrow{DA}.\overrightarrow{AD}+\overrightarrow{DA}.\overrightarrow{DC}+\overrightarrow{AB}^2+\overrightarrow{AB}.\overrightarrow{AD}+\overrightarrow{AB}.\overrightarrow{DC}=0\)
\(\Leftrightarrow0-h^2+0+a^2+0+ab=0\)
\(\Leftrightarrow a^2-h^2+ab=0\)
Đây là hệ thức cần tìm
