a) Ta có: BC song song với AD (ABCD là hình bình hành)
\(\Rightarrow\left\{{}\begin{matrix}M\widehat{C}N=A\widehat{D}C\left(đv\right)\\M\widehat{C}N=M\widehat{B}A\left(slt\right)\end{matrix}\right.\)
Xét \(\Delta MNC\) và \(\Delta AND\) Có:
\(\left\{{}\begin{matrix}\widehat{N}:chung\\M\widehat{C}N=A\widehat{D}C\left(cmt\right)\end{matrix}\right.\) \(\Rightarrow\Delta MNC\wr\Delta AND\left(g.g\right)\) (1)
Xét \(\Delta MNC\) và \(\Delta MAB\) có:
\(\left\{{}\begin{matrix}C\widehat{M}N=A\widehat{M}B\left(đ^2\right)\\M\widehat{C}N=M\widehat{B}A\left(cmt\right)\end{matrix}\right.\)\(\Rightarrow\Delta MNC\wr\Delta MAB\left(g.g\right)\) (2)
Từ (1);(2)\(\Rightarrow\Delta MNC\wr\Delta AND\wr\Delta MAB\) (đpcm)
b) Ta có: \(\Delta MNC\wr\Delta MAB\left(cmt\right)\)
\(\Rightarrow\dfrac{MC}{MB}=\dfrac{MN}{MA}\Rightarrow MC\cdot MA=MB\cdot MN\)
Vậy \(AM\cdot MC=BM\cdot MN\)
