.,i 2 3'
lsin2,4 sin2B sin2C
t-- l* v z Tir (1), (2) suy ra : tgA _tgB =tBC 123 = tEA .tsB .tsC _(Ee+Ea+tec\3 123[t+2+3) ^ tgA.tgB.tgc ( ry.e.tgn.tgC\3 - 6 =[.--- J
(vi tgA + tgB + tgC = tgA.tgB.tgC)
Suy ra i tgAlgB.tgc = I
(2)
(3)
(4)
Til (3), (4) suy ra: tgA=I, tgB=2, tgC=3 (5) t-
F
It- t-
I
-i
TiI (1), (2), (5) suy ra : sinZA = |
24
L
i sin2B = -;
m^ .nn .pP (m+n+ p)m+n+P
YOn ; Crin Tho zVd Qudc Bd Cdn,llAl, THPT chuyen Lf Tu Trong, TP Cdn Tho.
NGUYEN MINH HA
Bei T12l333. Cho tt? di€n ABCD ndi ilAp ntdt rndt cdu tatn O vd goi G ld trong tdrn ctia trt
di€n. Ldy didm M ndm ls€n trong hodc tr€n mdt cria hinh cdu dudng kinh OG. Cdc dudng thdng MA, MB, MC, MD cdt mdt ctiu tdm O ldn nira
tai cdc didm A1, Bt, Ct, D7 theo thft ta. Chti'ttg minh ritng 1,/(AtBp tD 1) > V(ABCD), rrong di ki hieul/ r'h[ the t{ch cria tti:di|n.
Ldi giii.'frudc hdt, ta neu Bd dd sau :
v(ABCD)
= *o -Y-u Y,'-Y'-- (l) v\ArBtCrDl) MAr.M4.MCt.Mq
Ndi dung Ud a6 nay chinh lh bhi tor{n Tl0l2S9
(TH&TT, 712001) ; vi viy xin khdng nhSc lai
chrlng minh ding thrlc (1) & day.
Gqi (0) lh mlt cdu dudng kinh OG ta c6 :
l9 *,<otl = MA'MA. = MB'MB. = MC'MC r = = MD.MDT= R2 - oM2 : MA MB MC MD ' MAt MBt MCr MDr MA2 +MBz +MC2 +MDz (2) R2 -oM2 Mat khr{c, ta c6 : MA2 + MB2 + MC2 + MD2 =
eA-ofr)'+foi-ofr1z +@e -oil)2 +
roo-ofrl'
= 4(R2+oM') - god.ofr = = 4(R2 -oMzysu6.tutd
L4i vi didm M nam trong hoac nam trOn mat
cdu (0) nco u6.uG < 0 (vi as(M6,@ < o do (Md, uG I >9oo), rhanh rhft :
MA2 + MB2 + MCz + MDz < 4(R2 - oM2) (3') Tir (2) vi (3'), r4p dung BDT vd trung binh
c6ng vi trung binh nhan, m duoc
.MAMBMCMD, \ , \ -r-f-f - MAt MBr MCr MDt Suy ra : MATMBTMC rMDl> MA.MB.MC.MD (5) Cudi cirng, ddi chidu (1) vn (5) ta thu duoc
BDT cdn tim : v(A1B{rDr)>v(ABCD)
Ding thrlc xhy rakhi vd chi khi Md.Md = O,
MA2 = MB2 = MC2 = MD2 hay M tring vdi tam O ctra mat cdu ngoai tidp trl di€,n ABCD.
NhAn x6t. l) TiI hc thfc (3) (dring vdi moi didm M
trong kh6ng gian, ta suy ra
MA2 + MB2 + MC2 + MD2 < q(fr - optz)
e pt6.vtG <o (3')
a <uo,vrGl=1frc >goo
e M thu6c hinh cdu d4c (Q) dudng kinh OG.
D6i chidu (2) vi (3') ta thu dugc kdt quai sau :
Qu! tich nhtng didm M trongkhOng gian sao cho
-MAMBMCMD
T- "" +"- + + _ <4 (4')
MAt M4 MCt MDr
ld hinh cdu ddc (holc hinh cdu d6ng) thm Q, duong kinh OG, bao gdm ci mf,t cdu (O) ln qu! tich nhtng didm M
saochol=4.
Day ld bni to6n qu! tich trong kh6ng gian, tuong tuv1i
bli to6n qu! tich trong met phing md chring ta dd gip
trongBdi todnT9ll33 (nhu ban Nguy'5n Van Hau, 11A1, THPT Y€n Dfrng, Bdc Giang di chi ra).
2)Ban Nguy€n C6ng Khanh, t t4 tgft Vinh Loc, Phri
LOc, Thla Thi6n - Hud vi mOt sd ban khiic cfrng chi ra rang neu M trirng vdi G thi ta gip lai bdi tor4n T101277.
vi dau ding thrlc xriy ra khi vh chi khi G tring v6i O, trlc ld khi vi chi khi ABCD ld m6t trl di0n gdn ddu. Hai ban
TrinhVdn Nam, l0Al, THPT chuy0n Phan BQi ChAu,
Nghd An vd, Nguydn Thdnh Cdng, 11A1, THPT Lf Trt
Trgng, TP Cdn Tho cdn nhAn x6t ring trong hinh hoc
phing dd c6 hai kdt quri tuong tr1 (khi thay t0 dicn ABCD
b&i tam gi6c ABC, v(ABCD) bdi diOn tich s(ABC). D6 li
hai bdi to6n di glp trong tap chf TH&TT :