Giải thích các bước giải:

a.Xét $\Delta ABN,\Delta ACM$ có:

Chung $\hat A$

$\widehat{ABN}=\widehat{ACM}$

$\to\Delta ABN\sim\Delta ACM(g.g)$

b.Từ câu a

$\to \dfrac{AN}{AM}=\dfrac{AB}{AC}$

$\to AN\cdot AC=AM\cdot AB$

c.Xét $\Delta AMN,\Delta ACB$ có:

Chung $\hat A$

$\dfrac{AM}{AC}=\dfrac{AN}{AB}$ vì $AN\cdot AC=AM\cdot AB$

$\to\Delta AMN\sim\Delta ACB(c.g.c)$

d.Xét $\Delta IBM,\Delta INC$ có:

$\widehat{MIB}=\widehat{NIC}$(đối đỉnh)

$\widehat{IBM}=\widehat{ABN}=\widehat{ACM}=\widehat{NCI}$

$\to\Delta IMB\sim\Delta INC(g.g)$

$\to \dfrac{IM}{IN}=\dfrac{IB}{IC}$

$\to\dfrac{IM}{IB}=\dfrac{IN}{IC}$

Mà $\widehat{MIN}=\widehat{BIC}$

$\to\Delta IMN\sim\Delta IBC(c.g.c)$

e.Ta có $\Delta IBM\sim\Delta ICN$ (câu d)

$\to \dfrac{BM}{CN}=\dfrac{IB}{IC}$

$\to \dfrac{BI}{BM}=\dfrac{CI}{CN}$

Vì $BD, CE$ là phân giác $\widehat{MBI},\widehat{NCI}$

$\to \dfrac{DI}{DM}=\dfrac{BI}{BM}=\dfrac{CI}{CN}=\dfrac{EI}{EN}$

$\to DE//MN$