Oe tn6y LOHP : A,OH'P' n€n PH : P'H',
suy ra AH + PH > AO + PH > AH' + P'H'.
Tt d6 dO t6ng AH + PH li: lcyn nhdt thi ta chi
cAn xdt vf tri di€mP tren cung nh6 G.
Cung dE thdy ringkhi tli6m P = C, ho[c khi di6m P = B, thi t6ng dQ dii clang xdt bing
2R < AH + PH. H<yn nta, khi didm P = A thi tdng dq ddi dang xdt bing 0.
T6m lpi, duoi ddy ta chi x6t di6m P nim trong cung nh6 CB sao cho t6ng OH + P/i.lcrnnhdt
vd lric d6 AH + PH : R + OH + PIl(h.1).
Thgc vfly, x6t di6m P nim trong cung nh6
Po,B thi tam gi6c OPPsc6 OP : OPo: R n6n
@ *Ga < l8oo,do d6dfu.soo:m, dfu.soo:m,
suy ra tia PoP nim gifia hai tia PoO vit PoDo ,
tuc ld di6m P nim b6n trong tam gi6c OPOo. Tir di6m P k6 dutrng thing PD // Ppovd PD
cilttia OB tqidiCm D thi di6m D nimgita hai di6m O vit Do, tuc ld OD < ODo, haY ld
oH+PH<OHs+HOo.
Chrmg minh tuong t.u khi tli€m P trung v6i
dii5m Pr nim trong cung nho flZ .
Trong tam gi6c vu6ng cdn OPOo thi cpnh
huy6n ODo: OPoJi: RJ2 .
Vay t6ng AH + PHl6nnfr6t khi di6m P trung
v6i giao diem Po cira dulng trdn vd tia OPo sao cho 664:45o. Lric d6 tong
AHs + PoIIo : R + OHo+ PoI-{o : R + RJ,
:R (1+O). o
DINH HUONG 2. Tren tia OC 6y hai di6m
K, E sao cho OK: PH vd KE - KP: OH vor tli6m K nim gifia hai di6m O vd E, hic t16
PH + OH: OK+ KE : OE, ddnd6n viQc x6t khi ndo thi doan thing OE ddinhdt.
Cdch gidi 2. Theo c6ch chon di6m E v6iKE : KP thltam gi6c PKE liltam gr6c vudng KE : KP thltam gi6c PKE liltam gr6c vudng cdn vd lu6n c6 @E : frF :45o (v6i c5c
diCm P, E b6tkl th6a mdn KE : KP). Kd tia
OPo cht nira tludng trdn tai di6m Po sao cho
^ ^
Eorn: eoen:45o. Ke P&o L oc, Polro L
OB (Ko e OC, Hs e OB) vi 16y di6m Eo tr6n tia OC sao cho K&o : PoKo v6i di6m Ko nim
gita hai di6m O vd Eo . Lric d6
OHo+ PoIIo: KrPo+ OI<o: OKs* KoEo: OEo vbi 6ffi: 6il4+ffirn: eoo , tuc ld
khi ducrng thing Pffi tiep xtc vdi ntra tlucrng
trdn (h.3).
Tuong tu nhu tr6n ta chimg minh tlugc n6u P nim trong cung nh6 6d mirP kh6c Pothi
OH+PH:OE<OEO
vd OHs+ PoIIo: OEo: OPoJi: RJ, .
Hinh3
K€t luqn: T0ng AH + PHllnntrAt ttri di6mP
tdng v6i Biao tlGm Ps cira du<rng trdn vd tia oPosao cho Eoro: 45o. Lric d6
AHs + PoIIo : R + OHo + Polfo : R + RJi
: R (r+'/z;. trDINH HUONG 3. St duns dinh li DINH HUONG 3. St duns dinh li
Pythagoras trong tam girlc vu6ng OHP litOfi i Prt: OP : R2 vd BDT zxy. *'* y' Ofi i Prt: OP : R2 vd BDT zxy. *'* y'
de d6nh gi6 t6ng (OH + rt42 tA*tdt.
Cdch giai 3. X6t di6m P nio d6 nim trongcung nh6 G . trot g tam gi6c vu0ng OHP c6 cung nh6 G . trot g tam gi6c vu0ng OHP c6
(oH + PIr': OI? + Prt + zOH. PH
: oP + 2oH. PH: Rz + 2oH. PH
< R2 + oI? + PI? :2R2,
titd6OH+PH<RJr,
styraAH*PH:AO+OH+PH
<R + RJ1:R (1+f ;.
Eing thric xity ra khi vd chi khi OH: PH ,
lric d6 tam gi6c OHP liL tam gr6c r,u6ng cdn,
tuc li di6m P trung vdi di€m Po vd di6m 11
trung v6i di6m 110 sao cho 604:45o. T6ng
OH + PH l6n ntr6t bang R 0+Jz) n6n t6ng
AH + PH lon nhSt bing
AHo+ PoIIo : R + OHo+ PoIIo : R + RJi
:R (r+'J7;.
K€t tuqn nhu tr0n.
,' olxu uIICINC 4. Sri dung BDT
Bunyakovsky vd dinh li Pythagoras trong tam gi6c vu6ng OHP ld Ort + Prt: OF : n2 a6
tlrinh gi6 tdng(OH + PH)216n1fi6t.
Cdch gidi 4. Xet di6m P ndo tl6 nim trongcung nh6 CB. Theo BET Bunyakovsky c6 cung nh6 CB. Theo BET Bunyakovsky c6
(oH+ PA':o.OH + t.PH)z
. (1'+ t1(ort + Prt) :zoP :2R2.
Tt d6 OH + PH < RJ, . Eing thfc xtty ra
khi vi chi khi OH : PH, l:
'c d6 tam gi6c
OHP liL tam.gi6c vudng c0n, tfc ld diOm P
tring v6i di6m Pe vd di6m 11trung v6i di6m
110 sao ctto -AOfn: 45o vdr c6
AHo + Porro :R + R Ji : n(l+o).
K€t luQn nhu tr6n. 0
EINH HUONG 5. Str dpng c6ng thric lugng gi6c trong tam gi6c vudng OPH.
Hinh +
Cdch gidi 4. Xet di6m P ndo d6 nim trongcung nh6 CB. Trong tam gi6c vu6ng OHP cung nh6 CB. Trong tam gi6c vu6ng OHP
v1ifiP:o<90oc6
OH + PH: OP cos cr * OP sin o
: R(cos o * sin ct) @.a). Chf i ring (cos o + sin o)2 : (cos2 o * sin2 o) + 2cos o. sin o
<1*cos2o*sin2a.:2.
Tn d6 OH + PH: R(cos o * sin u) < RJ2.Ding thric xtty ru khi vd chi khi OH: PH , Ding thric xtty ru khi vd chi khi OH: PH ,
lric d6 tam gi6c OHP lit tam gtdc vu6ng cAn,
tuc ld tli6m P tdng v6i tli6m Po vd di6m H
trrnrg vdi diCm Ho sao cho EO4:45o vd c6
AHo + Polro :R + R Ji : n(l+o).
K€t tudn nhu tr6n. D
DINH HU6NG 6. St dung hQ thtc lusngtrong tam gir{c vu6ng APB vbr g6c vu6ng 0 trong tam gir{c vu6ng APB vbr g6c vu6ng 0
dinh P vd BDT 2xy 1* * f AC Aann gi6 tdng (OH + rnf rcnnndt.
Cdch gidi 4. Xet di6m P ndo d6 nim trongcung nh6 CB cua nta dudng tron dulng kinh cung nh6 CB cua nta dudng tron dulng kinh AB. Ddt OH : x < R (h.4). Trong tam gi6c
4
vu6ng APB voi ZpB:90o vcvi ducrng cao PH
c6 hq thfc PHz: AH. HB : (Ao + oA(oB - HB): G + x) (R - "r) : R2-x2 , r_ hay PH : 1R" - x" Tt d6 (oH + PIl)' : 1* +.,[P -; 1' ^, :x2+R2-x2+zxtlR2- x2 ^ rn : R2 + 2xrln2 - *' < n2 + 1f + R2 - i):2R', suyraOH+PH<RJr.
Ding thric xhy rakhi vd chi khi OH: PH,ltc
d6 tam gi6c OHP ld tam gi6c vu6ng cAn, hic
.. -.t
ld tli0m P tnrng voi ili6m Po vd di6m I/ trung
vdi diOm I1o sao cho 6dFr: 45ovi c6
AHo + Polfo :R + R Ji : n 0+ Jr).
KAt hQn nhu tr6n. n
Hoan ngh6nh ban N4uye" Anh Tud:n, GY
tru<yng THCS Hoi Hi6u 2, TX Th6i Hod,
NghQ An,dd str dsng cdcbitding thric dC giai bdi to6n ndy, tuy nhi6n bpn bien dOi hoi dei.
Moi c6c b4n tim nhi6u c6ch gi6i cria BAITOAN 16 NCG (THCS) sau ddy vd giri vO TOAN 16 NCG (THCS) sau ddy vd giri vO
Tda sopn TH&TT trudc ngdy 30112/2014.
O BAI TOAN 16 NCG. Cho s6 daong a.
:,
L'ac so x, y lhoa mdn x -l y> a.
4
Chtrng ninh ring x4 + y4 > +
8
Edng thac xay ra khi ndo ?
HOANG NGTIYEN
(Nsh€ An)
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rlt
n
,
Unii:" dong ho dd tu lAu tro thdnh mQt vpt hiru dirng dic luc kh6ng th6 ttri6u dugc trong
cuQc s6ng cua m6i con ngudi. Vdi mQt quan niQm ph6 th6ng nh6t, moi nguoi trong chirng
ta, kh-6ng ai ld kh6ng bi0t su dung ddng hd
nhim rnpc dich nim virng vd di0u titit thcri gian bi6u cua c6 nh6n minh. Ngodi ra, d5i voi c6c
bpn hgc sinh, chiiic d6ng h6 vd ho4t d6ng cira
n6 ch[ra dlmg rAt nhi€u diAu b6 ich, thti vi vd
cflng gi6n dqn tqa vi6c xem xdt quy luAt hopt
dQng ctra chi5c d6ng h0 nhu mQt bdi to6n. Moi
c6c b4n cing t6i tim hiilu mQt sd bei to6n nho
quanh chi6c d6ng h6 cua chring ta.