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)
