cho tam giác ABC có AA'BB'CC' là đường cao CMR : a)\(\Delta ABC\sim\Delta A'B'C'\) b)\(AB'\times BC'\times CA'=AB\times BC\times CA\times\cos A\times\cos B\times\cos C\)
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
a: Xét ΔOAB và ΔOA'C có
OA=OA'
\(\hat{AOB}=\hat{A^{\prime}OC}\) (hai góc đối đỉnh)
OB=OC
Do đó: ΔOAB=ΔOA'C
=>AB=A'C
A'B=A'O+OB
AC=AO+OC
mà A'O=AO và OB=OC
nên A'B=AC
Xét ΔABC và ΔA'CB có
AB=A'C
BC chung
AC=A'B
Do đó: ΔABC=ΔA'CB
b: Xét ΔA'AC và ΔAA'B có
A'A chung
AC=A'B
A'C=AB
Do đó: ΔA'AC=ΔAA'B
c: Xét ΔOA'A và ΔOBC có
\(\frac{OA^{\prime}}{OB}=\frac{OA}{OC}\)
\(\hat{A^{\prime}OA}=\hat{BOC}\) (hai góc đối đỉnh)
Do đó: ΔOA'A~ΔOBC
=>\(\hat{OA^{\prime}A}=\hat{OBC}\)
mà hai góc này là hai góc ở vị trí so le trong
nên A'A//BC
a: Xét ΔOAB và ΔOA'C có
OA=OA'
\(\hat{AOB}=\hat{A^{\prime}OC}\) (hai góc đối đỉnh)
OB=OC
Do đó: ΔOAB=ΔOA'C
=>AB=A'C
A'B=A'O+OB
AC=AO+OC
mà A'O=AO và OB=OC
nên A'B=AC
Xét ΔABC và ΔA'CB có
AB=A'C
BC chung
AC=A'B
Do đó: ΔABC=ΔA'CB
b: Xét ΔA'AC và ΔAA'B có
A'A chung
AC=A'B
A'C=AB
Do đó: ΔA'AC=ΔAA'B
c: Xét ΔOA'A và ΔOBC có
\(\frac{OA^{\prime}}{OB}=\frac{OA}{OC}\)
\(\hat{A^{\prime}OA}=\hat{BOC}\) (hai góc đối đỉnh)
Do đó: ΔOA'A~ΔOBC
=>\(\hat{OA^{\prime}A}=\hat{OBC}\)
mà hai góc này là hai góc ở vị trí so le trong
nên A'A//BC