dinh li potoleme
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dinh ly potolemeTRANSCRIPT
chuyn :
B. nh l ptolemy.
nh l Ptolemy trong t gic:
a) Bt ng thc Ptolemy: Trong mt t gic, tng cc tch ca cc cnh i khng nh hn tch hai ng cho.
Cng thc:
Chng minh:
Cch chng minh quen thuc nht l:
Xc nh im M trong t gic sao cho v
Khi : suy ra BD.CM=AB.CD. (1)
T kt qu trn c:
suy ra BD.AM=AD.BC (2)
T (1),(2) c
BT c chng minh, du bng khi A,M,C thng hng hay t gic ABCD ni tip.
*Ngoi ra cn c nh l Casey Ptolemy tng qut, nhng trong bi vit ny, ti ch trnh by nhng ng dng ca Ptolemy cho t gic, cng l trng hp hay c s dng nht ca Ptolemy tng qut.
b) ng thc Ptolemy: Trong t gic ni tip, tng cc tch cc cnh i bng tch hai ng cho.
Cng thc:
y l trng hp c bit ca BT Ptolemy, khi t gic ABCD ni tip.C. mt s bi tp ng dng v m rng
I. ng dng chng minh mt s BT Hnh Hc v i S
Bi 1: (IMO Shortlist 1996):
Cho tam gic ABC ni tip (O) bn knh R. H,I,K theo th t l tm ng trn ngoi tip cc tam gic BOC,COA,AOB. Dng , , . Chng minh cc bt ng thc sau:
a) .
b) .
Chng minh:
t , , .
Khi , rt d dng, ta chng minh c:
v (1)
C t gic ABOC ni tip, theo ng thc Ptolemy: (2)
T (1) v (2) suy ra: .Tng t: v .
Nhn v vi v ca cc ng thc trn ta chng minh c cu a.
( Ch BT: vi x,y,z l cc s thc dng)
T kt qu trn ta cng chng minh c cu b.
Nhn xt: t cch chng minh bi ton trn ta c c bi ton tng qut hn cho IMOSL1996:
Bi 2: Cho tam gic ABC. O l mt im bt k nm trong tam gic. AO ct (BOC) A, BO ct (COA) B, CO ct (AOB) C. Chng minh cc BT:
a)
b)
Bi 3: Cho tam gic ABC v im M bt k nm trn (O) ngoi tip ABC. Chng minh rng:
Chng minh:Hin nhin ta c v phi.
Ta chng minh v tri bng phn chng: Gi s:
Khng gim tng qut, gi s M nm trn cung BC khng cha A
Do t gic ABMC ni tip, p dng ng thc Ptolemy c:
hay .
Khi :
(1)
Mt khc: suy ra:
v .
T :
(2)
(1),(2) mu thun, gi s l sai, vy ta c iu phi chng minh.
Bi 4: Cho t gic ABCD, AB//CD. Chng minh : .
Chng minh:D dng chng minh:
T BT tng ng vi BT: , y chnh l BT Ptolemy. Vy c pcm.
Bi 5: Chng minh BT sau vi a,b,c dng:
Chng minh: (hnh v trang sau)
Xc nh trn mt phng 4 im O,A,B,C sao cho v OA=a, OB=b, OC=c.
Dng nh l hm s cosin ta c cc kt qu:
, v .
S dng BT Ptolemy cho t gic OABC ta c:
Kt hp cc iu trn ta c iu phi chng minh.
*T bi ton trn ta c kt qu sau:
Vi x,y,z l cc s thc dng th ta c bt ng thc: (Singapore MO)
Bi 6 (INMO1998): Cho t gic ABCD nm trong (O) bn knh R = 1cm. Chng minh rng khi v ch khi ABCD l hnh vung.
Chng minh
Hin nhin ta c phn o.
Nu .
Ta c:
(V AC,BD c di khng vt qu 2.)
Du bng xy ra khi v ch khi t gic ABCD l hnh vung.
Vy ta c iu phi chng minh.
Bi 7: Cho tam gic ABC ni tip (O). phn gic ct (O) ti D khc A. K,L ln lt l hnh chiu ca B,C trn AD. Chng minh .
Chng minh:
Do BKA,CLA l cc tam gic vung nn ta c:
Mt khc t gic ABDC ni tip nn theo ng thc Ptolemy ta c:
. M ta c cng thc:
T suy ra: . Vy ta c pcm.
Ptolemy cng rt hiu qu trong vic xy dng v chng minh cc ng thc Hnh hc:
II. ng dng trong chng minh ng thc hnh hc:
Bi 8 (HongKong TST 2000) Cho tam gic ABC vung c BC>CA>AB. D l mt im trn cnh BC, E l mt im trn cnh AB. Ko di BA v pha A sao cho BD=BE=CA. Gi P l im nm trn cnh AC sao cho EBDP ni tip. Q l giao im th hai ca BP v (ABC). Chng minh: AQ+CQ=BP.
Chng minh:
D dng chng minh c: suy ra (1)
v (2).
Do t gic BEPD ni tip nn:
Vy c iu phi chng minh.
Bi 9 (China League 1989) Tam gic ABC (AB>AC) ni tip (O). Phn gic ngoi ti A ct (O) ti E. Gi F l hnh chiu ca E trn AB. Chng minh rng:
2AF = AB AC
Chng minh:
t . Bng cc cng thc lng gic ta c:
, , ,
Li c t gic AEBC ni tip, p dng ng thc Ptolemy ta c:
T suy ra: hay AB - AC = 2AF.
Vy ta c iu phi chng minh.
* Gi s vi trng hp c bit, AB = 3AC, ta c bi ton mi:
Cho tam gic ABC c AB = 3AC ni tip (O), phn gic ngoi ti A ct (O) ti E. F hnh chiu ca E trn AB. Chng minh tam gic AFC cn.
Bi 10 (China League 2000) Cho tam gic nhn ABC, im E,F trn cnh BC sao cho, dng v , ko di AE ct (O) ngoi tip tam gic ABC ti D. Chng minh .
Chng minh:
t , .
Ta c:
II- Mt s ng dng ca Ptolemy trong chng minh cc ng thc v bt ng thc trong tam gic
1. Xt tam gic ABC vi cc ng trung tuyn
Gi M,N,P ln lt l trung im ca BC,CA,AB.
- Xt hnh bnh hnh ANMP c : nn
Cng v cc BT tng t ta thu c kt qu:
1.1.
Lm tri bng BT Chebyshev ta c kt qu khc:
1.2.
- Ptolemy cho t gic APGN c:
, cng v cc BT tng t thu c kt qu:
1.3.
- Ptolemy cho t gic AHMN ta c:
, lm tri mt cht ta c kt qu sau:
1.4. ()
- Ptolemy cho t gic BPNC c ta li c kt qu khc:
1.5.
- Gi D,E,F l hnh chiu ca G trn CA,AB,BC. S dng Ptolemy cho t gic BEGF c . Khi ta c BT:
1.6.
2. Xt tam gic ABC vi cc ng cao
Gi AD,BE,CF l cc ng cao, H trc tm tam gic ABC.
- Ta thy t gic BFEC ni tip nn theo ng thc Ptolemy:
Cng theo v cc ng thc tng t ta c ng thc sau:
2.1.
- S dng Ptolemy cho t gic AEDF c:
, bin i 1 cht ta thu c BT sau:
2.2. . Du bng khi
Cng v cc BT tng t ta c:
2.3. (BT ny khng xy ra ng thc)
- Gi O l tm ng trn ngoi tip tam gic ABC; M,N,P theo th t l trung im ca BC,CA,AB. Hin nhin AH=2OM; BH=2ON; CH=2OP.
Khi vi cc t gic ni tip OMCN, OPBM, OPAN ta c ng thc:
2.4.
thay vo KQ trn ta c:
2.5.
- Do Ptolemy ng vi c t gic lm nn ta c BT:
2.6.
3. Xt tam gic ABC vi cc ng phn gic:
- Gi I,O theo th t l tm ng trn ni tip v ngoi tip tam gic ABC.
D,E,F theo th t l hnh chiu ca I trn AB,BC,CA.
S dng ng thc Ptolemy cho t gic AFIE c:
, tng t: v .
Ch vi IA,IB,IC ta u c: .Do I l tm t c ca h im (A,B,C) vi cc h s a,b,c. Suy ra: . Bnh phng 2 v ri rt gn ta c:
. Cng v cc BT ban u ta c kt qu:
3.1.
Ta c:
Thay vo BT 3.1 v kt hp vi 2.3 ta thu c 1 kt qu khc kh di, v khng c p mt cho lm
3.2.
C
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