\(a,\widehat{AMB}=\widehat{ANC}=90^0\\ \Rightarrow\Delta AMB\sim\Delta ANC\left(g.g\right)\\ \Rightarrow\dfrac{AM}{AN}=\dfrac{AB}{AC}\Rightarrow AM\cdot AC=AN\cdot AB\\ b,\cos BAM=\dfrac{AM}{AB}=\cos45^0=\dfrac{\sqrt{2}}{2}\)
Lại có \(\dfrac{AM}{AN}=\dfrac{AB}{AC}\Rightarrow\Delta AMN\sim\Delta ABC\left(c.g.c\right)\)
\(\Rightarrow\dfrac{AM}{AB}=\dfrac{MN}{BC}=\dfrac{\sqrt{2}}{2}=\dfrac{1}{\sqrt{2}}\\ \Rightarrow BC=\sqrt{2}MN\\ \Rightarrow BC^2=2MN^2\)
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