E.646.
{BM⋅BC=BD⋅BA=BE⋅BGCN⋅CB=CF⋅CA=CG⋅CE⇔{BM⋅BC=BE⋅(BE+EG)CN⋅CB=(CE−EG)⋅CE⇒ \begin{cases} BM \cdot BC = BD \cdot BA = BE \cdot BG \\ CN \cdot CB = CF \cdot CA = CG \cdot CE \end{cases} \Leftrightarrow \begin{cases} BM \cdot BC = BE \cdot (BE+\textcolor{red}{EG}) \\ CN \cdot CB = (CE-\textcolor{red}{EG}) \cdot CE \end{cases} \Rightarrow {BM⋅BC=BD⋅BA=BE⋅BGCN⋅CB=CF⋅CA=CG⋅CE⇔{BM⋅BC=BE⋅(BE+EG)CN⋅CB=(CE−EG)⋅CE⇒ ⇒BM⋅BCBE−BE=CE−CN⋅CBCE\Rightarrow \dfrac{BM \cdot BC}{BE} -BE = CE - \dfrac{CN \cdot CB}{CE}⇒BEBM⋅BC−BE=CE−CECN⋅CB
⇔BM⋅BCBE+CN⋅CBCE=BE+CE\Leftrightarrow \dfrac{BM \cdot \cancel{BC}}{BE} + \dfrac{CN \cdot \cancel{CB}}{CE} = \cancel{BE+CE}⇔BEBM⋅BC+CECN⋅CB=BE+CE
⇔BMBE=1−CNCE=NECE\Leftrightarrow \dfrac{BM}{BE} = 1- \dfrac{CN}{CE} = \dfrac{NE}{CE}⇔BEBM=1−CECN=CENE
⇔BMME=NENC.\Leftrightarrow \boxed{\dfrac{BM}{ME} = \dfrac{NE}{NC}}.⇔MEBM=NCNE.
