Cho tam giác ABC nhọn các đường cao AA', BB', CC' cắt nhau tại H.
CMR: \(\dfrac{HA'}{AA'}+\dfrac{HB'}{BB'}+\dfrac{HC'}{CC'}=1\)
K
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CM
31 tháng 12 2018
+ Các tam giác ABC và ABH có chung đáy AB nên tỉ số đường cao bằng tỉ số diện tích:
+ Tương tự:
Khi đó ta có
9 tháng 5 2022
a, Xét Δ ABD và Δ ABE, có :
\(\widehat{ADB}=\widehat{AEB}=90^o\)
\(\widehat{BAD}=\widehat{BAE}\) (góc chung)
=> Δ ABD ∾ Δ ABE (g.g)
b, Xét Δ EHB và Δ DHC, có :
\(\widehat{EHB}=\widehat{DHC}\) (đối đỉnh)
\(\widehat{HEB}=\widehat{HDC}=90^o\)
=> Δ EHB ∾ Δ DHC (g.g)
=> \(\dfrac{EH}{DH}=\dfrac{HB}{HC}\)
=> \(HB.HD=HC.HE\)
15 tháng 6 2022
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26 tháng 2 2018
a, Có : HA'/AA' = HA'.BC/AA'.BC = S AHB + S AHC / S ABC
Tương tự : HB'/BB' = S BHA + S BHC / S ABC ; HC'/CC' = S CHA + S CHB / S ABC
=> HA'/AA' + HB'/BB' + HC'/CC' = 2.(S AHC + S AHB + S BHC)/S ABC = 2
Tk mk nha
TH
7 tháng 4 2019
a)
'
AA
'
HA
BC
'.
AA
.
2
1
BC
'.
HA
.
2
1
S
S
ABC
HBC
; (0,5đi
ể
m)
Tương t
ự
:
'
CC
'
HC
S
S
ABC
HAB
;
'
BB
'
HB
S
S
ABC
HAC
(0,5đi
ể
m)
1
S
S
S
S
S
S
'
CC
'
HC
'
BB
'
HB
'
AA
'
HA
ABC
HAC
ABC
HAB
ABC
HBC
(0,5đi
ể
m)
b) Áp d
ụ
ng tính ch
ấ
t phân giác vào các tam giác ABC,
ABI, AIC:
AI
IC
MA
CM
;
BI
AI
NB
AN
;
AC
AB
IC
BI
(0,5đi
ể
m )
AM
.
IC
.
BN
CM
.
AN
.
BI
1
BI
IC
.
AC
AB
AI
IC
.
BI
AI
.
AC
AB
MA
CM
.
NB
AN
.
IC
BI
(0,5đi
ể
m )
12 tháng 3 2021
c) Bổ đề: Cho tam giác ABC có đường cao AH. Khi đó \(AH^2\le\dfrac{\left(AB+AC-CB\right)\left(AC+AB+BC\right)}{4}\).
Thật vậy, dựng hình chữ nhật AHCE. Lấy F đối xứng với C qua AF.
Ta có \(AH=CE=\dfrac{CF}{2}\).
Do đó \(CF^2+CB^2=BF^2\le\left(AB+AF\right)^2=\left(AB+AC\right)^2\Rightarrow CF^2\le\left(AB+AC-CB\right)\left(AC+AB+BC\right)\Rightarrow AH^2\le\dfrac{\left(AB+AC-CB\right)\left(AC+AB+BC\right)}{4}\).
Bổ đề được cm.
Áp dụng ta có \(\dfrac{\left(AB+BC+CA\right)^2}{AA'^2+BB'^2+CC'^2}\ge\dfrac{\left(AB+BC+CA\right)^2}{\dfrac{\left(AB+AC-CB\right)\left(AC+AB+BC\right)}{4}+\dfrac{\left(BC+BA-AC\right)\left(AC+AB+BC\right)}{4}+\dfrac{\left(BC+AC-AB\right)\left(AC+AB+BC\right)}{4}}=4\).
Vậy ta có đpcm.
12 tháng 3 2021
a) Ta có \(\dfrac{HA'}{AA'}=\dfrac{HA'.BC}{AA'.BC}=\dfrac{2S_{HBC}}{2S_{ABC}}=\dfrac{S_{HBC}}{S_{ABC}}\).
Tương tự \(\dfrac{HB'}{BB'}=\dfrac{S_{HCA}}{S_{ABC}};\dfrac{HC'}{CC'}=\dfrac{S_{HAB}}{S_{ABC}}\).
Do đó \(\dfrac{HA'}{AA'}+\dfrac{HB'}{BB'}+\dfrac{HC'}{CC'}=\dfrac{S_{HBC}+S_{HCA}+S_{HAB}}{S_{ABC}}=1\).
Ta có:
\(\dfrac{HA'}{AA'}+\dfrac{HB'}{BB'}+\dfrac{HC'}{CC'}\)
\(\dfrac{HA'.BC}{AA'.BC}+\dfrac{HB'.AC}{BB'.AC}+\dfrac{HC'.AB}{CC'.AB}\)
\(\dfrac{S_{BHC}}{S_{ABC}}+\dfrac{S_{AHC}}{S_{ABC}}+\dfrac{S_{AHB}}{S_{ABC}}=\dfrac{S_{ABC}}{S_{ABC}}=1\)