a, xet \(\Delta ABH\) va \(\Delta ACE\)
\(\widehat{A}chung\) ; \(\widehat{AHB}=\widehat{AEC}=90\)
\(\Rightarrow\Delta ABH\infty\Delta ACE\)
MINH SUA LAI DE CAU B:
\(\Delta CFA\) thanh \(\Delta AFC\)
b, xet \(\Delta AKD\) va \(\Delta AFC\)
\(\widehat{A}chung\) ; \(\widehat{AKD}=\widehat{AFC}=90\)
\(\Rightarrow\Delta AKD\infty\Delta AFC\)
c, xet \(\Delta ADK\) vuong tai K va \(\Delta CBH\) vuong tai H
CB = AD (ABCD la hinh binh hanh)
\(\widehat{HCB}=\widehat{DAK}\) ( so le trong)
\(\Rightarrow\Delta ADK=\Delta CBH\left(ch-gn\right)\)
\(\Rightarrow AK=CH\)
vi \(\Delta ABH\infty\Delta ACE\) => \(\dfrac{AB}{AC}=\dfrac{AH}{AE}\)
\(\Rightarrow AB.AE=AC.AH\)
\(\Rightarrow AB.AE=AC\left(AC-CH\right)\)
ma AK = CH (cmt)
\(\Rightarrow AB.AE=AC^2-AC.AK\left(1\right)\)
vi \(\Delta AKD\infty\Delta AFC\Rightarrow\dfrac{AD}{AC}=\dfrac{AK}{AF}\)
\(\Rightarrow AD.AF=AC.AK\left(2\right)\)
công (1) vao (2):
\(AB.AE+AD.AF=\)
\(AC^2-AC.AK+AC.AK\)
\(\Rightarrow AB.AE+AD.AF=AC^2\)
