diem torri cell i

7/26/2019 Diem Torri Cell i

1/9

Bi vit nh v im Fermat - Torricelli trongtam gic

Quang Long - Nguyn Minh Tun

im Torricelli hay cn gi l im Fermat l im m tng khong cch t im ti 3 cnh

l nh nht.

im ny l duy nht , n nhn cc cch ca tam gic di gc 120o , c mi lin h trc tip

vi bi ton Napoleon.

Trn hnh v l tam gic ABC, dng ra pha ngoi cc tam gic u ACD,BCF,BDA. Th

BD,CE,AE ng qui ti T l im Fermat ca tam gi ABC.

Trn hnh v ta thy 1 s tnh cht rt p nh :

1. Tam gic PMN u(nh l Napoleon)(M,N,P l tm cc tam gic u )

2. BD=CE=AF; AM,BN,CP ng quy ( ti im Napoleon)

3. Tam gic ABC v tam gic DEF cng trng tm. . . .

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d i e n d a n t o a n h o c . n e t [ V M F ]

D F a i d . C o m # 1 P d f S o l u t i o n s
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2/9

Mt vi tnh cht khc :

1) Gi s BD ct ng trn ngoi tip tam gic BEF D. Tng t c E,F. Th ta s

c BD = BD + AF + CE. ( hnh v pha trn )

2) Ly trn AF, BD, CE, cc im A,B,C sao cho AA

AF + BB

BD + CC

CE = 1 th tamgic ABC u

Ta i vo mt s tnh cht ca im Fermat :

Tnh cht 1 :Cho tam gic ABC, T l im Torricelli ca tam gic . CMR ng thng i

xng vi AT qua BC, BT qua CA , CT qua AB ng qui.

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3/9

Hng dn :Ly A,B,C l im i xng ca T qua BC, CA, AB. Ta s chng minh ng

thng i xng vi AT qua BC, BT qua CA , CT qua AB ng qui ti O l tm ng trn

ngoi tip tam gic ABC.

Tht vy gi s ng thng i xng vi BT qua CA , CT qua AB ct nhau ti O. Dotnh cht i xng qua BC nn OB A = 180o ATB = 60o. Tng t c ACB = 60o. MAC = AT = AB ACB = ABC. Do OCB = OB C. M CAB = 2. BAC nnCOB = 240o 2.ATa c : BAC= BTC= 120o, CBA = 2. CAB, ACB = 2. BCA. Nn CAB = 120o A.Nn BOC = 2. CAB. Do O l tm ngoi tip tam gic ABC.

Tnh cht 2 : ng thng Euler ca Tam gic ABT, BCT, CTA ng qui.

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4/9

Hng dn

Gi G l trng tm ta gic ABC.Dng tam gic ACD u.

Gi O1 l tm ca tam gic u.

M l trung im AC, J l trng tm

ta gic ATC.

Ta c BT i qua C. ATCD nt tip

(O1)

JO1 chnh l ng trng Euler ca

tam gic ATC.

J v O1 chia MT v MD theo t l

1:2 Nn O1Ji qua G

Do ng thng Euler ca tam

gic ATC i qua trng tm tam gic

ABC

Tng t cho ng thng Euler ca

BTC, BTA cng i qua G nn chng

ng quy.

Tnh cht 3 : im Fermat v im Isodymatic trong l 2 im lin hp ng gic.

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5/9

Ta gii thch 1 s khi nim:

+) ng trn A-apollonian l ng trn i qua 3 im A,D,D. Vi AD v AD ln lt l

phn gic trong v ngoi gc A.

+) im Isodymatic trong l giao im ca 3 ng trn A-apollonian , B-apollonian, C-

apollonian v nm trong tam gic k hin l J.Tng t ta cng c im Isodymatic ngoi l giao im ca 3 ng trn A-apollonian ,

B-apollonian, C-apollonian v nm trong tam gic k hin l J.

+) 2 im I v J ca tam gic ABC l lin hp ng gic nu tia AI v AJ i xng vi nhau

qua phn gic trong ca gc A. Tng t vi BI,BJ; CI v CJ

Ta quay vo bi ton .

Ta cn 1 b cng l tnh cht kh p ca im Isodymatic trong.

B 1 : Gi M,N,P ln lt l hnh

chiu ca J xung BC,CA,AB. Khng

my kh khn ta x cm c tam gic

MNP u.

B 2: T b 1 v tnh cht ca im Fermat T ta c :BIC+

BJC= 180

o

+BAC.

Tht vy Ta c BJC= BPM+ MNC= 180o APM+ 180o ANM= 60o + BAC.BTC= 120o nn BIC+ BJC= 180o + BAC.

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6/9

Ta cn thm 1 tnh cht kh hu ch na : im I v J ca tam gic ABC l lin hp

ng gic ca tam gic ABC khi v ch khi : BIC+ BJC = 180o + BAC, CIA+ CJA =

180o + ABC, AIB+ AJB = 180o + BCA. T nhn xt ny v b 2 ta c pcm .

Tnh cht 4 :Khng kh khn ta s chng minh c tnh cht p sau:T l im fermat ca tam gic ABC. A,B,C l im nm chnh gia cung BTC, ATC, BTA

ca cc ng trn ngoi tip tam gic BTC, ATC, BTA. CMR

a) Tam gic ABC u

b) Gi O1, O2, O3l tm ng trn ngoi tip tam gic ATB,TCA,TAB. th tam gic O1O2O3

u, gi O l tm tam gic . Khi O cng l tm tam gic ABC

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7/9

Tnh cht 5 : Vi iu kin cc gc

ca tam gic nh hn 120o th tamgic ABC c T l im Fermat trong,

tng t im Fermat ngoi bng cch

dng tam gic u cng pha vi A, B,

C nh trong hnh v.

Ta c AA1,BB1,CC1 ng quy ti T.

T v T c nhiu mi tnh cht p s gii thiu sau y:

a) Gi M,N,P l trung im ca BC,CA,Abv K l trung im ca TT th M,N,P,K

cng thuc 1 ng trn. (K thuc ng

trn Euler ca tam gic ABC )

b) Gi G l trng tm ca Tam gic ABC, I l im lin hp ng gic ca G. Gi H l trc

tm tam gic ABC. S l trung im GH. Th ta c T,T,S,I thng hng

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8/9

c) (nh l Lester) : Cho tam gic ABC khng cn, O l tm ng trn ngoi tip tam gic ,

T v T l 2 im Fermat, I l tm ng trn Euler ca tam gic ( ng trn 9 im ). CMR

O,T,T,I thng hng.

Tnh cht 6 : Mt s ng thc v bt ng thc v im Fermat- Torricelli.

Cho tam gic ABC cc gc nh hn 120o, T l im Fermat trong, T l im Fermat ngoi.

a)(TA.TB+ TB.TC+ TC.TA)(TA + TB+ TC) =TA.a2 + TB.b2 + TC.c2

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9/9

b)(TA + TB+ TC)2

3(BC.PA + CA.PB+ AB.PC)

c) Gi s G l trng tm, M l im bt k trong tam gic. Th th

MA3 + MB3 + MC3

MG2 +

a2 + b2 + c2

9

(TA + TB+ TC)

d) Nu S l im Isodymatic ni trn, da l khong cch t S ti phn gic trong

gc BAC, tng t c db, dc. Th th

SA + SB + SC=F A + FB+ FC+ 2

d2

a

SA+

d2b

SB+

d2c

SC

e) 1

TA+

1

TB+

1

TC 3

RviRl bn knh ng trn ni tip tam gic.

f) cos2(A)

TA +

cos2(B)

TB +

cos2(C)

TC cos(A) + cos(B) + cos(C)

2R

g) Gi O l tm ng trn ngoi tip. Th th

SOTT = 1

3. |(a2 b2)(b2 c2)(c2 a2)|

(a2 b2)2 + (b2 c2)2 + (c2 a2)2

h) Gi G l trng tm tam gic , O xc nh nh trn . CMR GO2 + GT2 + GT2 = R2

khi R l bn knh ng trn ngoi tip tam gic.

Mt s bi tp khc

1. Cho tam gic ABC vi gc BAC= 60o , T l im Fermat- Torricelli, M l trung im BC.

CMR TA + TB + TC = 2.AM .

2. Cho tam gic ABC vi gc BAC= 60o , T l im Fermat- Torricelli, D nm tia i tia CT

sao cho tam gic ATD u . Trung trc TA ct BC O. CMR QT l trung trc BD .

3. Nu TG= min{GA;GB;GC} =GA ( G l trng tm tam gic ). CMR BAC= 120o.

4. Cho tam gic ABC, T l im Fermat. Dng ra pha ngoi tam gic cc tam gic u

BCA1,CAB1,BAC1. Gi sAA1 ct ng trn ngoi tip tam gic A1B1C1 ti A. Tng

t c B v C. Gi sA2 l im i xng ca A qua A. Tng t c B2, C2. Gi O l tm

ng trn ngoi tip tam gic ABC, O l tm ng trn ngoi tip tam gic A2B2C2. CMR

O,O,T thng hng.

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