tl'ttlr try ld dLrong phdn giac trong cua cdc tam
gidc OBC, OCA, OAB. Goi OAz, OB2, OC2
theo thu ta ld dudng phdn giitc trong cua cdc tam giac OAA;, OBB;,-OCC1.- Chtrng minh rdng
( +)'. [++'l' . ( *\' >(z * ll)'
\ArA,) \BrB,) \CrCt) '
Ding thirc xiry ra khi ndo? Loi Sitii. (Theo nhidu ban).
D4tOA=a,OB=b,
OC = c, BOC= 4
De thar, .0, - &*c' 2'2-2: B, = Ct*A ., c, =4i!
Suy ra arbrc2 = 8r3 sinAz sinBz sinCz B, +C; C1+41 A1 +81 = 6r'' Sln Slfl- Sl
222
> 13(sinB1 +sinC1)(sinG +sinAr)(sinAr +sin&)> ,'.ZJrinA,t*C,.2 rlrrC, rr"a, . ZG;A, *re > ,'.ZJrinA,t*C,.2 rlrrC, rr"a, . ZG;A, *re = Sr3sinArsin&sinCr = arbrcr. Suy ra
^3 L3 ^3
++ ) a tb ( r = 8r3 sinAr sinBr sinCr .
ai Dici
: IU do vol cnu y rang
B+C-C+A^A+B,4,=-'.8,=-iC,=-,ta'222 ,4,=-'.8,=-iC,=-,ta'222 a1blc1 A B C " ' )Brr cos-cos-cos- 6dA= p, eos = r. . Tac6 a+P+Y = 18ff. co Theo tinh chdt
dudng phAn gi6c 4
trong tam gi6cOAA, ta thtiy OAA, ta thtiy
AAz OA a
AzAt OAt OAr suy
AA, AAZ , A
AzAt AzAr OAt
athcr 222
8rr ^ 8ra a
abc abc
Til d6 v6i chf 1i rang p> 3J1r, ta cd
a)b)c3 ,21616
athcr abc
Ding thrlc xiy ra 'a LA.BC ddu. O
.B&lUOns. tUco lUOns. tUco -=l+BtBt bCCt.c -lT- O& CzCr OCr
Dat vd trdi ciua bdt ding thrlc cdn chrlng minh
le P thi theo BDT Bunhiacovski:
rBnN $,ffie* qirdi$s s6 35814-2007) 25i qirdi$s s6 35814-2007) 25i p(p - a) bc p(p-c) ab
( o\'(. b)'P=ll+_ I +l l+ _-l P=ll+_ I +l l+ _-l ' l. oAt) \ oh) t( a b .) >-l?+ +-+-i - l'[" oAr oh ocr ) , \2 ( c ) +l 1+- I \ oc') z
Mdt kh6c, 6,P dung BDT Cauchy tajg
a,b-'r.,r@Q)
oh' o&- oq ' '! oa, 'oBt'ocl
Sft dung cOng thrlc tinh dO dii dudng phdn
gi6c trong cta tam gric,tathu duoc
NguydnThi Hanh Dung, 12 To6n' THPT chuy0n Hir
ff"rfr,-Sf.h OinhVd XudnThanh,1246' THPT 56 2'
Tuy Ph,rO.; Si niu - Ving Tdu: Dinh Ngoc Thdi'
ii'foan, THPT chuven Lc Quf Don, TP'-Ving Tdu; il-Org Nir, TrdnThi Huvdn, 1l To6n I' TIT chuvdn
iloris ita virtt' vintt Longt Clyqnr Cdns Danh'l lTr,iHt'f chuy6n Nguy6n Binh Khiem' l lTr,iHt'f chuy6n Nguy6n Binh Khiem'
Hd Qu,qNc vmn *gal Llt354. -\ior nqrdf dang chay vcti vdn
tdc i trAn mdt tlLit iit:iii til6[ hr)n da vcti vdt't
t6c ban ddu i so rri; )''J.tt':; tlii Bi|t tt : t''
hdy tim Soc ll hctp hti: "' :! "'(i'i phLr(rtig
n[org di tinr nint x't t'i' ' :: '-' ]'':ot.ll cttLt
n'a, Ea kt lot'r nhdt. Bo,ir'; ".- ";': ::iiin:1 kJti r) chiirt cLt() ngtroi.
(1)
OAt.OBt.OCr--
Do (a+bXb+ c)(c+a) > Sabc vd Lni gitii. Chon hq
truc toa dO OxY nhu
hinh v6. Gsi io l,vAn tdc ban ddu ctra vAn tdc ban ddu ctra
hbn dd so v6i mat
ddt, a lh g6c hoP boi
io vdi mat ddt. Tdm
xa cira hdn dr{ theo
phuong ngang Ii xo = v& *inza '
I
Ta c5 v-6 =i +it. Theo dinh li sin:
sinP sin(P-a) sinaVi v = u nan B = 2a,suy ra Yo = 2r"cosa' Vi v = u nan B = 2a,suy ra Yo = 2r"cosa'
8v2
Do vAy xu= L.cos3a.sina
o A
Theo BDT CauchY, ta lai c6
111.
1 =lcos2 a +lcosZ a +-cos- a L stn- a
J33>4 >4 1 -'-= J: (3) ap COS-. COS-
1) ."or' 28='f , non til (3) suY ra
abc ,
oA.O&.OCt
(4)
8
JVJ
Til (1), (2) vd (4) ta nhAn duoc
,
-:2
p=l[:*3i/+ I = 1z+^.,61'
3[ \3J3/
Ding thric xiY ra khi vh chi khi