EM THAM KHẢO : 

$\textbf{a)}$

$\text{Xét } \Delta BDA \text{ và } \Delta BFC:$

$\widehat{B} \text{ chung}$

$\widehat{BDA} = \widehat{BFC} = 90^\circ$

$\Rightarrow \Delta BDA \sim \Delta BFC \text{ (g-g)}$

$\Rightarrow \dfrac{BD}{BF} = \dfrac{BA}{BC}$

$\Rightarrow BD \cdot BC = BF \cdot BA$

$\textbf{b)}$

$\text{Từ } \dfrac{BD}{BF} = \dfrac{BA}{BC} \Rightarrow \dfrac{BD}{BA} = \dfrac{BF}{BC}$

$\text{Xét } \Delta BDF \text{ và } \Delta BAC:$

$\widehat{B} \text{ chung}$

$\dfrac{BD}{BA} = \dfrac{BF}{BC}$

$\Rightarrow \Delta BDF \sim \Delta BAC \text{ (c-g-c)}$

$\Rightarrow \widehat{BDF} = \widehat{BAC}$

$\textbf{c)}$

$\text{Xét } \Delta BDH \text{ và } \Delta BEC:$

$\widehat{B} \text{ chung}$

$\widehat{BDH} = \widehat{BEC} = 90^\circ$

$\Rightarrow \Delta BDH \sim \Delta BEC \text{ (g-g)}$

$\Rightarrow \dfrac{BD}{BE} = \dfrac{BH}{BC}$

$\Rightarrow BH \cdot BE = BD \cdot BC$

$\text{Xét } \Delta CDH \text{ và } \Delta CFB:$

$\widehat{C} \text{ chung}$

$\widehat{CDH} = \widehat{CFB} = 90^\circ$

$\Rightarrow \Delta CDH \sim \Delta CFB \text{ (g-g)}$

$\Rightarrow \dfrac{CD}{CF} = \dfrac{CH}{CB}$

$\Rightarrow CH \cdot CF = CD \cdot CB$

$\Rightarrow BH \cdot BE + CH \cdot CF = BD \cdot BC + CD \cdot CB$

$\Leftrightarrow BH \cdot BE + CH \cdot CF = BC(BD + CD)$

$\Leftrightarrow BH \cdot BE + CH \cdot CF = BC \cdot BC = BC^2$

$\textbf{d)}$

$\text{Do } AM \parallel BC \Rightarrow AM \parallel BD$

$\Rightarrow \Delta AFM \sim \Delta BFD \text{ (Hệ quả định lý Thales)}$

$\Rightarrow \dfrac{AM}{BD} = \dfrac{AF}{BF}$

$\Rightarrow AM = \dfrac{AF \cdot BD}{BF}$

$\text{Từ câu a, } \Delta BDA \sim \Delta BFC \Rightarrow \dfrac{BD}{BF} = \dfrac{AD}{CF}$

$\Rightarrow AM = AF \cdot \dfrac{AD}{CF} = \dfrac{AF \cdot AD}{CF}$

$\text{Xét } \Delta ADC \text{ và } \Delta BEC:$

$\widehat{C} \text{ chung}$

$\widehat{ADC} = \widehat{BEC} = 90^\circ$

$\Rightarrow \Delta ADC \sim \Delta BEC \text{ (g-g)}$

$\Rightarrow \dfrac{CD}{CE} = \dfrac{AD}{BE}$

$\Rightarrow CD = CE \cdot \dfrac{AD}{BE} = \dfrac{CE \cdot AD}{BE}$

$\text{Xét tỉ số: } \dfrac{AM}{CD} = \dfrac{\dfrac{AF \cdot AD}{CF}}{\dfrac{CE \cdot AD}{BE}} = \dfrac{AF \cdot BE}{CF \cdot CE}$

$\text{Xét } \Delta ABE \text{ và } \Delta ACF:$

$\widehat{A} \text{ chung}$

$\widehat{AEB} = \widehat{AFC} = 90^\circ$

$\Rightarrow \Delta ABE \sim \Delta ACF \text{ (g-g)}$

$\Rightarrow \dfrac{AB}{AC} = \dfrac{BE}{CF} = \dfrac{AE}{AF}$

$\Rightarrow AE \cdot CF = AF \cdot BE$

$\Rightarrow \dfrac{AM}{CD} = \dfrac{AE \cdot CF}{CF \cdot CE} = \dfrac{AE}{CE}$

$\text{Do } AM \parallel CD, \text{áp dụng hệ quả định lý Thales } \Delta ICD:$

$\dfrac{AI}{ID} = \dfrac{AM}{CD}$

$\Rightarrow \dfrac{AI}{ID} = \dfrac{AE}{CE}$

$\text{Trong } \Delta ADC, \text{ có } I \in AD, E \in AC \text{ và } \dfrac{AI}{ID} = \dfrac{AE}{CE}$

$\Rightarrow IE \parallel DC \text{ (Định lý Thales đảo)}$

$\Rightarrow IE \parallel BC$