a/ Xét \(ΔAEB\) và \(ΔAFC\):

\(\widehat A:chung\)

\(\widehat{AEB}=\widehat{AFC}(=90^\circ)\)

\(→ΔAEB\backsim ΔAFC(g-g)\)

\(→\dfrac{AE}{AB}=\dfrac{AF}{AC}\)

\(↔AE.AC=AF.AB\)

b/ \(\dfrac{AE}{AB}=\dfrac{AF}{AC}\)

\(↔\dfrac{AE}{AF}=\dfrac{AB}{AC}\)

Xét \(ΔAEF\) và \(ΔABC\):

\(\widehat A:chung\)

\(\dfrac{AE}{AF}=\dfrac{AB}{AC}(cmt)\)

\(→ΔAEF\backsim ΔABC(g-g)\)

\(→\widehat{AEF}=\widehat{ABC}\) 

c/ Xét \(ΔHFB\) và \(ΔHEC\):

\(\widehat{HFB}=\widehat{HEC}(=90^\circ)\)

\(\widehat{FHB}=\widehat{EHC}\) (đối đỉnh)

\(→ΔHFB\backsim ΔHEC(g-g)\)

\(→\dfrac{HF}{HB}=\dfrac{HE}{HC}\)

\(↔\dfrac{HF}{HE}=\dfrac{HB}{HC}\)

Xét \(ΔHFE\) và \(ΔHBC\):

\(\dfrac{HF}{HE}=\dfrac{HB}{HC}(cmt)\)

\(\widehat{FHE}=\widehat{BHC}(=90^\circ)\)

\(→ΔHFE\backsim ΔHBC(g-g)\)

\(→\widehat{HEF}=\widehat{HCB}\) (1)

Xét \(ΔCEB\) và \(ΔCDA\):

\(\widehat{CEB}=\widehat{CDA}(=90^\circ)\)

\(\widehat{C}:chung\)

\(→ΔCEB\backsim ΔCDA(g-g)\)

\(→\dfrac{CE}{CB}=\dfrac{CD}{CA}\)

\(↔\dfrac{CE}{CD}=\dfrac{CB}{CA}\)

Xét \(ΔCED\) và \(ΔCBA\):

\(\dfrac{CE}{CD}=\dfrac{CB}{CA}(cmt)\)

\(\widehat C:chung\)

\(→ΔCED\backsim ΔCBA(g-g)\)

\(→\widehat{CED}=\widehat{CBA}\)

hay \(\widehat{CED}=\widehat{CBF}\)

Ta có: \(\widehat{FCB}+\widehat{CBF}=90^\circ,\widehat{CED}+\widehat{BED}=90^\circ\)

mà \(\widehat{CBF}=\widehat{CED}\)

\(→\widehat{FCB}=\widehat{BED}\) hay \(\widehat{HCB}=\widehat{HEN}\) (2)

(1)(2) \(→\widehat{HEF}=\widehat{HEN}\)

\(→EH\) là đường phân giác \(\widehat{FEN}\)

\(→\dfrac{HN}{NE}=\dfrac{HF}{EF}\) (3)

Xét \(ΔNEH\) và \(ΔNCD\):

\(\widehat{HNE}=\widehat{DNC}\) (đối đỉnh)

\(\widehat{NEH}=\widehat{NCD}(cmt)\)

\(→ΔNEH\backsim ΔNCD\)

\(→\dfrac{HN}{NE}=\dfrac{DN}{CN}\) (4)

(3)(4) \(→\dfrac{HF}{EF}=\dfrac{DN}{CN}\)

\(↔DN.EF=HF.CN\) (ĐPCM)