a) Kẻ MH ⊥ AC ( H thuộc AC )
+ ΔAMH nửa đều \(\Rightarrow AM=2AH\)
+ ΔMHN vg tại H \(\Rightarrow MN^2=MH^2+NH^2=AM^2-AH^2+NH^2\)
\(=x^2+\left(AH^2+HN^2\right)-2AH^2=x^2+\left(AH+NH\right)^2-2AH^2-2AH\cdot NH\)
\(=x^2+y^2-2AH\left(AH+NH\right)=x^2+y^2-xy\)
b) \(\frac{AM}{BM}+\frac{AN}{CN}=1\Leftrightarrow\frac{x}{a-x}+\frac{y}{a-y}=1\)
\(\Leftrightarrow x\left(a-y\right)+y\left(a-x\right)=\left(a-x\right)\left(a-y\right)\)
\(\Leftrightarrow ax+ay-2xy=a^2-ax-ay+xy\Leftrightarrow a^2-2ax-2ay+3xy=0\)
\(\Leftrightarrow x^2+y^2-xy=a^2+x^2+y^2-2ax-2ay+2xy=\left(a-x-y\right)^2\)
\(\Leftrightarrow MN=a-x-y\)
