Gia su' hai Irung luycn BM va CN ^-^
cat nhau tai G. , , . , . )( ii mud .ji.iii ijiioul oi/nJ ; Ta C O : . ,
colA = 2(colB + cotC) 4S
, o b" + c-^ - a" = 4a" ..^
c:>b^^ + c- = 5a\)
Mat khac: BM 1 CN o BC' = B G ' + C G '
a
<=> a" = ~*f"h + ."^1;)
<=> a' = — , 4
o b^ + c' - 5a'. (2)
T i r C l ) (2) suy ra dpcm.
Bai 4. Goi G la Irong tarn tam giac ABC va gia sir AGB 4 C = 180".
ChiJng minh: a' + b" = 2c". ' ii - j d i o —
, _ Giai ''-'--k '
Do AGB4-C 180"
=> col AGB = - c o t C ' J . ; .
G A - + G B " AB- a ^ + b ' - c ^ / ^ ^ ^ ^ ^ 4S
S
4S
SCAB =^ T G A " 4 G B' = ^( m ^ + )
3
a" 4- b" + 4c
a^ + b 2 +4c^
ncn thay vao (1) ta c6:
- C " 2 2 , 2
c — a — b
4S 3 4S
'J
a' + b" = 2c" => dpcm.
Cty rmn nrv DVVH Khang VIft
jjai 5- ^''"^ S'a*-" ABC thoa man he ihifc:
sinA + sinB + sinC = 2 . A . B . Cl sin— 2 2 2 f sin —sin — ChiTngmmh: — - ^ - + - .
I, h c TiT gia thiol, ta c6:
?fO'J
A B C ^ 4 cos — cos — cos — = 2 2 2 2 cos -
Giai
B + C . B . C) 9 . 9 9 + sin—sin — a 4- A . A B C ^
4cos — cos — c o s — 2
2 2 ^ , B C A ' " B " C ^ " ^ ' O 2cos —cos —cos—=- cos —cos — 2 2 2 2 2
B C . B . C . B C cos —cos sin—sin— 1 sin—sin —
An2 2 2 2 2 2 j
A 1
O COS — = — 9 9
dt) cos —B C C O S — > 0 9 9
fllr-
I) i i i i u b <]/
ol„ = be
b 4-c
2 be cos A do 1., b ^ c 1 1 1
O — — f -
L, b e
dpcm.
Nhqn xet: Bai loan trcn thiTc chat c6 dang:
ChiTng minh rang trong tam giac ABC la c6:
sinA + sinB + sinC = 2 A . B C sin h9 9 9 sin—sin — 1 1 I
<^ — = - + - .
'^ai 6. ChiJng minh rang trong lam giac ABC, la c6:
A B
a + b = c + he lan — 4- lan — ^^-^ 1.
Ta c6: a 4- b = c + h, <=> -a + b - c 2 2 Giai
= 1
2R(sinA4-sinB~sinC) _ " " , . (f, f~ v., , - , , 2S acsinB
vw da sir dung cong ihu^c h^. = — = a sinB' 1=-- asm B) 2S acsiii'6':
2R(sin A +sinB -sinC) _, c e
2Rsin AsinB = 1 - A I :ri(!iii! anljriO
^ . A + B A - B - . C C 2 sin cos - 2 sin ^ cos
2 2 2 2 , . A . B A B 4 sin sin cos cos
2 2 2 2
= 1
cos A - B A + B
cos —COS
2 2 cos
^ . A . B A B 2 sin sin cos cos
2 2 2 2
= 1 C : > 2
. A + B
sin A D
0 , A B ,
<=> 1 o tan — + tan — = 1 A B 2 2 cos C O S
2 2
A B cos COS -
2 2
d p c m .
= 1
B a i 7. C h o l a m g i a c A B C i h o a m a n he ihuTc: + _ ^ ' " ^ s i n 2 A + s i n 2 B s i n 2 C ChuTng m i n h : c o s A + c o s B = 1.
G i a i A p d u n g t i n h c h a t c i i a l i Ic ihiVc, la c 6 :
s i n C sin A + sin B sin A t s i n B + s i n C s i n 2 C s i n 2 A 1 s i n 2 B s i n 2 A + s i n 2 B + sin^ C
T i r i D c o :
A B C
4 c o s cos cos 2 2 2 4 s i n A s i n B s i n C
1
^ . A . B , C Ssin sin sin
2 2 2 ( 1 )
2 c o s C ^ . A . B , C 8sin sin sin
-) -) T
„ , . A . B . C
• cosC = 4 s i n — sin — sin — -) T T
•cosC = 2 A B
cos- cos-A + B l
n ^ A . B
• cosC - 2 cos cos 2 A B
sin — . C
2 sin — 2 2 2
=> c o s C = c o s A + c o s B - 1 + cosC
=> cos A + c o s B = I 1 ^ d p c m .
B a i 8. C h o l a m g i a c A B C i h o a m a n d i c i i k i c n sin A + sin B + s i n C
ChuTng m i n h :
cos A + cos B + cosC
= 0 . ( T^] A - - B - 71 ^ ( T^) C - -
l 3 J 3 . l 3 J
G i a i Tiif gia i h i c t , la c 6 :
( s i n A - N/3 COSA) + ( s i n B - S c o s B ) + ( s i n C - N/3 COSC) = 0
1 73
— sin A cos A 2 2 O S i n
CJ. 2 sin
+ sin B + sin C - -
3 3 3
— S i n B cos B 2 2
= 0
+ - s i n C c o s C 1 . ^ V3
2 2 = 0
A + B T\
2 ~1
A - B
cos + s i n C - ^
3 = 0 . ( 1 )
, „ A + B 7t T T - C TX TX C . ,
^—2—I = ^~T^6"7'""^"'^^'^'^"^
(1) <=> 2sin 1 6 ^ 2
A B ^ . cos 1- 2 sin
cos TT
2 6 2 6, = 0
<=> sin ( c 7T A - B cos — - cos
6 2 6 ; 2 = 0
— ^ov
nod
<=>
sin
cos
C _ TT 2 6 ) C _ T T1 2 6
= 0
= cos-A - B
3
C TT _ A - B 2 6 ^ 2 C T T _ B - A 2 6 ^ 2 C - - = - - 0
3
C + B - A = - o 3 C 4 A - B = ^
3
C - - = () 3 A - ^ = . 0
3 B - ^ = . ( )
3 rCr d o siiy ra
3 B 3
A - ^
3 = 0 .
',n'0'lJl l i ( U . U t l ;il(/(l-l i J l M i l i
fO Mi
B a i 9.
C h o l a m g i i i c A B C t h o a m a n he thuTc: s i n 5 A + s i n 5 B + s i n 5 C = 0. ih u Chu'ng m i n h r a n g l a m g i a c c 6 it n h a t m o t g 6 c bang 3 6 " h o i l c 108". A
^- C h o t a m g i a c A B C t h o a m a n he ihtfc s i n 3 A + s i n 3 B + s i n 3 C - 0.
ChuTng m i n h r h n g l a m g i a c c6 it n h a t m o t g o c bang 6 0 " h o a c 120". -y- G i a i
T a c 6 : s i n 5 A + s i n 5 B + s i n 5 C = 0 : r i 1:1 •• •
• Eolduanit h^i S1HHgiOl LUjUiggiUL - /I I M I iiuy niMar
. 5A 4 5B 5 A - 5 B , . 5C 5C
^ 2sin cds h2sin cos () . (1)
1,1
V i
5A f 58
sin sin 90" 5C
C t ) S -5C
(2) d
Thay (2) vao (1) va c6:
5C 5A - 58 5C 5A + 53 •
cos — cos ^ cos — cos U
2 2 2 2 a + A
Ci- cos-5C 5 A - 58 5A + 5B cos 1 cos
j i ; ; i r
= 0.
Co hai kha nang sau xay ra:
5C
1. N c u cos — 0 T
. ^ ti A
Do 0 < ^ < 450", ncn tiY (3) c6:
1
^ 5A + 5B 2. N c u cos h cos
5A - 58 2 5C
< >
270"
36"
C 108"
. -'^A 58
O 2 cos cos — " 0 <^ A - - 3 6 " hoac A 108"
8 ^^-^ 36" hoac B 108".
Do la dpcm.
2. Chiang minh hoan loan lifdng l i f nhu" phan 1/.
Hai 10. Cho lam giac A B C ihoa man d i c u kiOn (1 + c()tA)( 1 + cotB) = 2.
Chi'fng minh: C = 45". ^ ( l i a i
j ') ^i^ iti/. bh "uT Tit gia I h i c l la c6: 1 + cot A + colB + ct)tAcotB = 2
=> cotA + cotB + colAcotB = 1 ^
^ colAcotB - 1 = - (colA + cotB). (1) M'tt N h i r d a b i c l liH)ng m o i A A B C , la c6: c o l A + colB ^ 0, nC-n liT (1) C O M I B ] oil'')
ct)t A c o i B - J j j j f u ; ^ "diy^nirl -JCVJ )om ihii'. ly mnl %nh rinirn y n t i f D col A + col B ... •)r.nix + 8F,im + /J.im -finU Ofi O H A -jiiiy, ffuu or!')
=> coKa + B)P TiLii "()r) ^nfld J O I J Jofii JH3i) ii 6J -jBia ffifiJ y n ^ i rtnini gntirlD Do 0 < A + B < 180", ncn tif (2) c6: j i j i ; j
A + B = 135" => C = 45" => dpcm. ; a i u + cif.ni.'. + /-Jjwr: :oj B1
" • • ..'< . Cty TIVtH nrv DVVnKhang Vlft
IJai 11- Cho tarn giac A B C c6 d i c u k i c n S = a' - ( b ' - c").
8
Chifns: m i n h rang: tan A ^ — ' • ' 15
G i a i
A p dung dinh l i ham so'cosin, ta c6 tu" gia ihic't:
— he si n A ^- h" + c" - 2 he cos A - b" - c" + 2 be
>
. - b c s i n A = 2be(l - cosA) (-f
/ I f . - (••'':
. . A A „ . . A
• be sin — cos— 4bcsin — i>..,\ / i 'ni
2 2 2 •
: d y >y
cos— 4sm— d o s i n — 0 2 2 2 A 1
=> Ian— - — 2 4
A 1 2 tan
=> tanA = ^ = — ~ = — => dpcm.
1 - t a n ^ ^ 1 - ' '-'^
2 16
IJai 12. Cho tam giac A B C thoa man d i c u k i c n nr, = e Chi?ng m i n h :
1. sinA = 2sin(B - C ) , 2. cotC = 3eolB.
G i a i 1. Tir m,, = e m,^ = e^
2 b 2 + 2 e 2 - - a ^ ,
^ ^ = c-
= > 2 ( b - c ^ ) = a^ (*)
=> 2(sin'B - sin-C) = s i n ' A ^ "
=:> (1 - cos2B) ~ (1 - eos2C) sin^A ^ l i - '
=> co.s2C - cos2B = sin"A
= > - 2 s i n ( C + B ) s i n ( C - B ) = s i n ' A . ( I )
Do sin(B + C) = sinA > 0, ncn liT (1) suy ra sinA = 2sin(B - C) => dpcm.
2- T i r ( * ) i a c 6 :
a= + b ' - c - = 3 ( a ' + c - - b-)
_^ a + b- - e a + e- - b ^ ^ ^
=> — 3——— => cotC = 3eotB => dpcm. ^ •
BAI du6ng Itpc abih 0ol LUpng gac - nan nuy lUtal
Nhdn xet:
1. X e t each g i a i khac sau day cho phan 21.
V i 111;, = c => A M = A B ( d day A M la di/dng trung t u y c n kc iCr A ) , ncn B A M la tarn giac can di'nh A .
Ke A H 1 B M ^ H M = H B . (2) A Ta c6:
^ HC . „ HB cotC = v a c o t B =
A H A H TCP (2) suy ra HC = 3 H B ,
nen c6 ngay cotC = 3cotB => dpcm. H M 2. Phan 2 cua bai loan la triTdng hdp rieng ciia bai loan sau:
Cho tarn giac A B C v d i B > C. V e trung tuyen A M . D a t A M B = a.
ChuTng m i n h : 2cota = cotC - cotB.
Giai
2
Ta c6: cot a - (do S = 2SABM)
A M ^ + B M ^ - A B ^ + - c 4S ABM
2 b ^ + 2 c ^ - a ^ 2 + c 4 4 2S
2S
Tir do: 2 cot a = 2b^ ~-2C^ a ^ + b ^ - c ^ 4S a H c ^ - b ^
= cot C - c o t B
4S 4 S 4S
Khi A M = A B (tiJc m„ = c), ta c6 a = B , nen tiir tren suy ra 2cotB = cotC - cotB => cotC = 3cotB
Nhan x e t dUrtc chiJng minh.
3. TiTcJng tiT la c6 bai loan sau: Cho lam giac A B C khong phai la lam giac vuong va thoa m a n he Ihiirc m , = R. ChuTng m i n h rSng:
l a n A l a n B = — . 3 TCf m , = R
=> 2sin^A + 2sin^B - sin^C = 1
1 - cos2A + 1 - cos2B - sin^C = 1
=> cos^C - 2cos(A + B)cos(A - B ) = 0
=> cos^C + 2cosCcos(A - B ) = 0 cosC[cosC + 2cos(A - B ) l = 0
Do cosC ^ 0 ncn suy ra: cosC + 2cos(A - B) = 0
=> -co.s(A + B ) + 2cos(A - B ) = 0
- c o s A c o s B + sinAsinB + 2cosAcosB + 2sinAsinB = 0
=> 3sinAsinB = - c o s A c o s B f .u.t 4 , Do cosAcosB * 0, la d i den dpcm: lanAtanB = - i . v ) h i l ; - ' . Bai 1 3 . Cho tam giac A B C v d i I , O ti/dng tfng la l a m di/dng tron n o i tiep va
jg ^ J
ngoai tiep tam giac. Gia sur sin—sin—sin — = .
2 2 2 4 V i , \\
Giai 1:
A B C I '
A p dung cong thu'c r = 4Rsin—sin—sin —, nen ttj" gia thiet ta c6: ' '
2 2 2 ,vi
r . A . B . C N/ 2 - 1 ':i
• ~ sin—sin —sin— -
4R 2 2 2 4 .,
= > r + R = R V 2 i i > f . ;
=> r ' + 2Rr + R ' = 2 R ' i:-).,-^
=>r^ = R - - 2 R r . ( I ) Theo cong thijfc Euler ta c6: R" - 2Rr = 1 0 ' , vay tiT (2) suy ra:
IO = r = > d p c m .
Bai 14. Cho lam giac A B C khong vuong l a i A, tai B vii thoa m a n d i e u k i e n : o2 , u2 2 ^r>2 • , l a n A l a n B + l
a + b - c = 4R . Chu'ng m m h : =: tan C . tan A tan B - 1
Giai Ta c6: a" + b^ - c' = 4R"
<=> s i n ' A + s i n ' B - sin'C = 1
O s i n ' A + sin'B = 1 + sin'C i A : .;
o 1 - C O S2A + 1 - C O S2B = 2 + 2sin^C O - c o s ( A + B)co.s(A - B) = sin-C
<=> c o . s C c o s ( A - B ) = .sin'C. (1)
Tir gia ihiet suy ra a' + b ' - c' > 0 a' + b ' > c^ => C la g6c nhon vay cosC > 0. K e t hdp v d i sin(A + B ) = sinC > 0, nen sau k h i chia ca hai ve cua (1) cho cosCsin(A + B ) , ta c6:
cos(A - B) • •
s i n ( A + B) < ' J , i . . .
^ cos A cos B + s i n A sin B „ " '
^ = tanC ,
sin A cos B + sin B cos A , t . ,
Bvi uuuiiy tun sum yiot Luyng yuc - nmn iiujrTmaT 1 + t a n A t a n B -
=> = t a n C . (2) tan A + Ian B
M a i k h i i c la c 6 : l a n A + l a n B = t a n C ( t a n A l a n B - 1) ( 3 ) (diCa v a o l a n A + l a n B + l a n C = l a n A l a n B l a n C ) .
, , I r t a n A t a n B ^
T i r (2) (3) d i d e n ; ^ l a n C P.O.J tan C( l a n A l a n B 1)
I + l a n A l a n B • ' ^ 4 * '
=> ^ I a n " C d p c m . Ian A l a n B - 1
Nhqii xet: G i i i i h i c l l a m g i i i c A B C k h o n g v u o n g l a i A h o a c B la c a n i h i c l . V i gia sii- n c u A = 9()" ^ a = 2R => a ' + b" - c' = 4R' + b" - c'
L u c n a y n c u c h o n A B C la l a m g i a c v u o n g c a n l a i A . K h i do l o r a n g i h o a m a n d i c u k i c n :
a- + b- - c- = 4R-.
T u y v a y l u c n a y l a n A k h o n g c6 nghTa, v l i h c k c t l u a n c i i a b a i l o a n la k h o n g d u n g .
B a i 15. C h o l a m g i a c A B C i h o a m a n d i c u k i c n : l a n A l a n C = 3 va l a n B l a n C = 6.
ChiVng m i n h r a n g k h i do la c 6 : l a n C = l a n A + l a n B . C i a i
T i J - l a n A l a n C = 3 =^ l a n C ( 1 ) tan A
lan B lan C = 6 -4> lan B — ^ . (2) l a n C
TCr (1) (2) suy r a l a n B = 2 l a n A . (3) A p d u n g c o n g ihi'fc
, „ , , 2 lan A -f ^
„ _ l a n B l a n C , . t;,n A
l a n A = - t a n ( B + C ) = hay l a n A = ^ i U l i l l a n B lan C i 5
=> 5 l a n ' A = 2 l a n * A + 3
= 5 t a n' A = l . (4)
TCr g i i i i h i c l suy r a l a n A , l a n B , l a n C c u n g d a u v a d o d o no p h i i i c i i n g du'dng ( v i n c u l a n A , l a n B , l a n C c u n g a m i h l A , B, C d c u la goc i t i m ; u i i h u a n vt1i A + B + C = 1 8 { ) " )
V i I h c lijf (1) c 6 : l a n A = 1 l a n C = 3 vii l a n B = 2.
N o i r i e n g la c 6 : l a n C = l a n A + l a n B => d p c m . Chu y: Vt'Ji b a i l o a n n a y , k h o n g c6 m c n h dc d i i o
T h a i v a y x c l b a g o c A , B , C e ( 0 ; 180") sao cho l a n A - l a n B = ^ 2 (=> l a n A + l a n B = l^jl ) va l a n C = 2>/2
N h i r v a y ro r a n g la c 6 : l a n A + l a n B = l a n C , , , tan A ' lan B , , ,
M a i k h a c : — — - uui( A > B) va t h c o each xac d i n h i h i : lan A lan B I
tan A ' l a n B
- 2 ^ 2 ^ l a n C . U i n A l a n B - l V 2 . V 2 - i
T i Y d o l a n C = - t a n ( A + B ) = : > A + B + C = 180".
N h L f l h c A , B , C la ba goc c i i a m o l l a m g i a c .
T r o n g l a m g i a c nay ro r a n g : l a n A l a n C = -Jl.l-Jl 4 3 , >' • ) (•'' l a n B l a n C = 4 ;t 6.
N o i each k h a c tiT:
l a n B + l a n A = l a n C k h o n g i h c suy ra l a n A l a n C = 3; l a n B l a n C = 6, li?c la m c n h dc d a o n o i c h u n g k h o n g d u n g .
B a i 16. C h o l a m g i a c A B C i h o a m a n d i c u k i c n (a + b f c ) .
Chu'ng m i n h : " l a - — a mu - m.. c ' 2
= 0 .
G i a i .11 >
T i r giii I h i c l la c 6 : m., + mn + m , = — a + — b + — c .
2 2 2
Ta c 6 :
( 1 )
2 ^ - <2b- + 2c2 - a^) + (2a- + 2c- - b^) + (2a^ + 2b~ - c ' )
3(a- ^ b- -f C - )
is
— a
2 1
[V3 ]
is
— a f- — b + — c
2 2 +
2 ( 2 ) f\
u
T i r (1) (2) suy ra:
i l , n i h + n%m, + m , m , = - 2
IS S, S ' — a t b I c 3 2
+ + S 2 C
^ S. ^f3 sf3 S V3 7 3 ,
— b. c t c. a H a. b .
h [not r/i " \
I \ i : f : :
; (3)
a S i d u a n g h^ x tinh g i o l I Ufing giac - rTtan nuy nnai
Binh phiTOng hai vc ciia (3), ta c6:
mlml + mlml + m^m.^ + 2mj,mbm^.(m^ + nib + m^.)
9. 2 2 , .2,.2 , ..2u2^ . 9
= ^ ( b \ - ^ + c ' a - + a - b " ) + -abc(a + b4c). (*) 16 8 Mat khac ap dung cong thiJc tinh diTcJng trung tuyen (sau khi rut gpn) ta c6:
16