Геометрия7_9
.pdf8 KJiaC |
80 |
8
B
A
D
MaJI. 116 |
Mu. 117 |
"lloTHPHKYTHHK noaHaqaIOTL, aanHcyroqH :Horo BepmHHH. HanpHKJlaµ;, 'IOTHPHKYTHHKHa MaJIIOHKY 117 noaHaqaioTL TaK: ABCD. y Il03Ha'leHiti'IOTHPHKYTHHKaBepmHHH, :w;o CTOSITL nopSIA, IlOBHHHi 6yTH cyci,zv1iMH. "lloTHpHKYTHHK ABCD Ha MaJIIOHKY 11 7 Mm1rna TaKom noaHa'IHTHBCDA a6o DCBA. AJ1e He MomHa noaHaqHTH ABDC (B i D - He cyci,D;Hi BepmHHH).
CyMa .zi;oBmHH ycix cropiH 'IOTHPHKYTHHKaHa3HBaGTLCS: nepuMerpoM.
51. IlAPAJIEJIOrPAM
llapa;n,eJtozpaM - ~e 'IOTHPHKYTHHK,y SIKOro npOTHJiemHi CTOpOHH napaJieJILHi, To6To J1emaTL Ha napaJ1eJ1LHHX npa:MHX (MaJI.
118).
6.1. HrcUfO iJiazona1ii 1'0TUpurcyrHurca neperuHa10TbC1' i 6 roii14i neperuHy iJi1'1'TbC1' nonollaM, ro 14eit 1'0TupuxyrHU1' - napa1ieJiOZJKUI.
Ao B e A e H Ha. Hexaii ABCD - AaHHH 110THPHKYTHHK i 0 - TO'IKanepeTHHY :Horo AiaroHaJie:H (MaJI. 119).
TpHKYTHHKH AOD i COB piBHi. Y HHX KYTH npH BepmHHi 0 piBHi HK BepTHKaJILHi, a OD = OB i OA = OC aa yMOBOIO TeopeMH.
0Tme, KYTH OBC i ODA piBHi. A BOHH e BHyTpimHiMH piaHo-
A
MaJI. 118 |
MaJI. 119 |
§ 6. qOTHpKKYTHHKH |
81 |
cTopouuiMH npH np.sMHX AD i BC i ci11uiii BD. 3a 03HRKOIO napaJieJibHOCTi np.sMHX np.sMi AD i BC napaJieJibHi. TaK caMo AOBOAHMO napaJieJibHiCT:& npSIMHX AB i CD aa ,11;on0Moro10 pieuocTi TPHKYTHHKie AOB i COD.
OcKiJibKH npoTHJiemHi cTopoHH 'IOTHpHKYTHHKanapaJieJlbHi, To 3a 03ua11eHHSIM ~eii 110THPHKYTHHK - napaJieJiorpaM. TeopeMy ,11;oee,11;euo.
52. BJIACTHBICTb ,ll;IAl'OHAJIEB IIAPAJIEJIOrPAMA
Teo p e Ma 6.2 (o6epueua ,11;0 TeopeMH 6.1). ,l(iazo1141ii TUJ.pa1ie- 1'0ZpaM.a nepe'rUH4Hn''bC1' i ti TO"l.1fi neperuHy iJi1'1'T'bC1' non.OJI.a.Al.
Ao e e A e H H .s. Hexaii ABCD - p;aHHii napaJieJiorpaM (MaJI. 120). Ilpoeep;eMo iioro p;iarouaJib BD. Ilooua11HMO ii: cepeAHHY 0 i ua npop;oemeuui eip;pi3Ka AO Bi,zi;KJia,D;eMo eip;piaoK OCi. ~o ,11;opieu10e AO.
3a TeopeMoIO 6.1 'IOTHPHKYTHHKABC1D e napaJieJiorpaM. 0Tme, np.sMa BC1 napaJieJibHa AD. AJie 11epe3 TO'IKYB Momua npoeeCTH TiJibKH op;uy np.sMy, napaJieJibHY AD. A ~e o3ua11ae, ~o np.sMa BC1 36iraeTbCSI 3 np.sMOIO BC.
Auanori11Ho AOBOAHMO, ~o np.sMa DC1 36iraeTbCSI 3 np.sMoIO DC. 0Tme, TO'IKaC1 36iraeTbCSI 3 TO'IKOIOC. IlapaJieJiorpaM ABCD 36iraeTbCSI 3ABC1D. ToMy iioro p;iaroHaJii nepeTHHaIOTbCSI i B TO'I~
nepeTHHY ,D;iJISITbCSI nonoJiaM. TeopeMy p;oeep;eHo. |
|
|||
3 a A a 11 a |
(6). "tJepe3 TO'IKYnepeTHHY ,D;iaroHaJieii napa- |
|||
JieJI~rpa_Ma npoeep;~uo np.sMy. ,D;oee,11;iTb, ~o ei,D;p~30K li, |
||||
SIKHH MlCTHTbCSI Ml.)I(. napaJieJibHHMH CTOpOHRMH, |
,ll;lJIHTbCSI |
|||
e ~ii TO'I~ nonoJiaM. |
|
|
|
|
P o 3 e'SI a a H H .s. |
Hexaii |
ABCD - ,11;aHHH |
napaJieJio- |
|
rpaM i EF - |
np.sMa, |
~o nepeTHHae napaJieJlbHi cropoHH |
||
AB i CD (MaJI. 121). TpHKYTHHKH OAE i OCF pieHi 3a ,11;pyro10 |
||||
03HaKOIO. Y HHX cTopoHH OA i |
OC pieui, 60 0 - |
cepep;HHa |
||
,11;iarouaJii AC. KyTH npH eepmHHi 0 pieui SIK eepTHKaJibHi,
Cr |
|
-=;;;...__~_,_.. c |
[ |
|
A D A
MaJI. 120 |
MaJI. 121 |
8 KJiac |
82 |
a KYTH EAO i FCO piBHi HK BHyTpimHi piaHOCTopoHHi npH |
|
napaJieJibHHX AB, CD i |
ci•rniH AC. |
3 piBHOCTi TpHKYTHHKiB BHilJIHBa6 piBHiCTb CTOpiH: |
|
OE= OF, ~o ii Tpe6a |
6yJio ,n;oBeCTH. |
53. BJIACTHBICTb IIPOTHJIE»tHHX CTOPIH
I KYTIB IIAPAJIEJIOrPAMA
Te ope M a 6.3. Y napan,en,ozpa;wa npoTuJ£e:HCni cToponu pisni, npoTUM:HCni KYTU pisni.
,ZJ; o B e A e H Ha. Hexaii ABCD - ,n;aHHH napaJieJiorpaM (MaJI. 122). IlpoBe,n;eMo .n;iaroHaJii napaJieJiorpaMa. Hexali 0 - ToqKa Ix nepeTHHY.
PiBHiCTb npoTHJie>KHHx cropiH AB i CD BHilJIHBa6 a piBHOCTi TPHKYTHHKiB AOB i COD. Y HHX KYTH npH BepmHHi 0 piBHi HK BepTHKRJib'Hi,a OA = OC i OB = OD aa BJiaCTHBiCTIO ,n;iaroHaJieH napaJieJiorpaMa. TaK caMo a piBHOCTi TPHKYTHHKiB AOD i COB BHilJIHBae piBHiCTb ,n;pyroI napH npoTHJie>KHHX cTopiH - AD i BC.
MaJI. 122 |
MaJI . 123 |
PiBHiCTb npoTHJie>KHHx KyTiB ABC i CDA BHnJIHBa6 a piBHOCTi TPHKYTHHKiB ABC i CDA (aa TpboMa CTOpoHaMH). Y HHX AB = CD i BC = DA aa ,n;oBe,n;eHHM, a cTopoHa AC cniJibHa. TaK caMo piBHiCTb npoTHJie>KHHX KYTiB BCD i DAB BHilJIHBa6 a piBHOCTi TpHKyTHHKiB BCD i DAB. TeopeMy ,n;oBe,n;eHo.
3 a A a q a (18). p;~Be~iT~, ~o K0JIH B qoTHpHKYTHHKY ,n;Bi
0 |
napaJienorpaM. |
CTOpOHH napaJieJibHl H plBHl, TO BlH - |
|
Po a B'a a a H Ha. Hexali ABCD - |
,n;aHHH qoTHpHKYT- |
HHK, y .HKoro cTopoHH AB i CD napaJieJibHi i piBHi (MaJI. IlpoBe,n;eMo qepea BepmHHY B npaMy b, napaJieJibHY cTopoHi AD. I.J;a npaMa nepeTHHae npoMiHb DC y ,n;e.HKili To~i C 1• "lloTHPHKYTHHK ABC1D 6 napaJieJiorpaM. OcKiJibKH y napaJieJIOrpaMa npoTHJie>KHi cTopoHH piBHi, TO C 1D = AB. A aa YMOBOIO AB = CD. 0Tme, DC = DC 1. 3Bi,n;cH BHilJIHBa6, ~o ToqKH C i C 1 a6ira10TbCS1. TaKHM qHHOM, qoTHpHKYTHHK ABCD a6iraGTbCH a napaJieJiorpaMOM ABC 1D, a n;e oaHaqa6, ~o BiH Tem 6 napaJieJiorpaM.
§ 6. qOTHpHKYTHHKH |
83 |
|
54. IIPHMOKYTHHK |
|
llp.11.MmCyTHUlC - |
:u;e rrapaJieJJorpaM, y .RKOro BCi KYTH |
rrp.RMi |
(MaJI. 124). |
|
|
T e o p e M a 6.4. ,lfiaz01ia11.i npaMoKyTHUM piBHi. |
|
|
,lJ; o Be /1. e H H .R. |
Hexaii ABCD - 11.aHHH rrp.RMOKYTHHK |
(MaJI. |
125). |
|
|
TBep11.meHH.R TeopeMH BHIIJJHBae a piBHOCTi rrp.RMOKYTHHX TpHKYTHHKiB BAD i CDA. Y HHX KYTH BAD i CDA rrp.RMi, KaTeT AD crriJILHHii, a KaTeTH AB i CD piBHi .RK rrpoTHJiemHi cropoHH rrapaJieJiorpaMa. 3 piBHOCTi TPHKYTHHKiB BHIIJIHB86, ~o ix rirroTeHy3H piBHi. A rirroTeHyaH e 11.iaroHaJI.RMH rrp.RMOKYTHHKa. TeopeMy .n;oBe11.eHo.
c
Man. 124 |
Man. 125 |
3 a /1. a q a (24). ,lJ;oBeAiTL, |
~o |
c
K~)JI~ y rra~aJieJiorpaMa uci KYTH |
|
||
plBHl, TO BlH 6 IIp.RMOKYTHHKOM. |
|
||
Po a B'.Ra a H H .R. |
KyTH |
rrapa- |
|
JieJiorpaMa, rrpuJierJii |
11.0 |
011.Hiei |
|
CTOpOHH, 6 BHYTpimHiMH 0/1,HOCTO- |
|
||
pOHHiMH (MaJI. 126), TOMy ix cyMa |
|
||
11.opiBHIOe 180°. OcKiJILKH aa YMO- |
Man. 126 |
||
BOIO aa11.aqi ~ KYTH pieHi, TO |
|
||
KOmH~ii a HHX rrp.RMHii. A rrapaJieJiorpaM, |
y .RKoro Bci KYTH |
||
IIp.RMl, 6 IIP.RMOKYTHHKOM. |
|
|
|
55. POMB
PoM.6 - :u;e rrapaJieJiorpa·M, y SKOI'O sci CTOPOHH piBHi (MaJI.
127).
T e o p e M a 6.5. ,lfiaz<11UUi poM6a MJH!TUHa.10Tw:11. nio n.pS&MuM KYTOM. ,lfiazoHa.11.i f10ltl6a e 6ice1"1'puca111u iioio JCyriB.
,n; o Be /1. e H H .R. Hexaii ABCD - 11.aBHi: poM6 (MaJI. 127), 0 - TOl!Ka rrepeTHHY iioro 11.iaroHaJieii. Sa BJI&CTHBiCTIO napaJieJiorpaMa AO = OC. 0Tme, y TpHKYTHHKY ABC Bi/l,piaoK BO e Me/l,iaHoIO.
8 KJIRC |
84 |
8
D
Man. 127
OcKinbKH ABCD - poM6, TO AB = BC i TpHKYTHHK ABC - pisuo6eApeHHii. 3a BJiaCTHBiCTIO piBHOOeApeHoro TpHKYTHHKa MeAiaHa, npOBeAeHa AO iioro OCHOBH, 6 6iceKTPHCOIO i BHCOTOIO. A ~e oauaqae, ~o Aiaroua.nb BD e 6iceKTpucoIO KyTa Bi nepneHAHKyiu1pua AO AiaroHaJii AC. TeopeMy AOBeAeHo.
3 a A a 'Ia (33). ,lJ;oBeAiTb, ~o KOJIH y napaneJiorpaMa |
|
AiarouaJii nepneHAHKYJIHpHi, TO BiH 6 poM6oM. |
|
Po as'a a a H Ha. Hexaii ABCD - |
napaJienorpaM, Aia- |
roHani HKOro nepneHAHKYJIHpHi, i 0 - |
TO'IKanepeTHHY Aiaro- |
HaJieii (MaJI. 128). TpHKYTHHKH AOB i AOD piBui aa nepmo10 03HaKOIO piBHOCTi TpHKYTHHKiB. y HHX KYTH npH BepmHld 0 aa YMOBOIO npaMi, CTOpoHa AO cniJibHa, a OB = OD aa
A
Man. 128
sJiacTHBicTIO Aiaroua.neii napaJienorpaMa. 3 piBHOCTi TpHKyTHHKiB sunnHBae pisuicTb cTopiu AB = AD. A aa BJiaCTHBicTIO npoTHJie>KHHX cTopiH napanenorpaMa AD = BC, AB = CD. C>rme, sci CTOpoHH napa.ne.norpaMa piBHi, To6To siH e poM6oM.
56. KB~AT
KeaiJpar - ~e npHMOKYTHHK, y HKOro BCi CTOpOHH piBHi (MaJI. 129). OcKiJibKH CTOPOHH KBaApaTa pisui, To siu e poM6oM. ToMy KB8ApaT Mae BJiaCTHBOCTi npaMOKYTHHKa i poM6a:
§ 6. qOTHpHKyTHHKH |
85 |
MaJI. 129 |
MaJI. 130 |
1.Y 1C8a01H"Ta Bci ICYTU npR.Mi.
2.,lfiazoHtVii 1C84iJl""T4 piBHi.
3.,lfiazoHtMi 1C8aiJl""ra neperu"HaK>Tbcsi niiJ npsiMUM ICYTOM i e 6ice1CTpuca.Mu 4ozo 1CyTi8.
3 a A a 'Ia (40). ,ZJ;oBe/I,iTh, ~o KOJIH 11.iaroHaJii npSIMOKYTHHKa nepeTHHaIOTbCSI ni11. npSIMHM KYTOM, TO BiH 6 KBa/I,paTOM.
Po a B'SI a a H H SI. OcKiJILKH npSIMOKYTHHK e napaJieJiorpaM, a napaJieJiorpaM a nepneH/I,HKYJISIPHHMH /I,iaroHaJISIMH
6 poM6 (aa/I,a'la 33), TO y poarJISl/I,YBaHoro npSIMOKYTHHKa BCi CTOpOHH piBHi (MaJI. 130). 3a 03Ha'leHHSIMTaKHH npSIMOKYTHHK - KBa/I,paT.
57. TEOPEMA tz,AJIECA
Teo p e Ma 6.6 (TeopeMa <l>aJieca). HICUfO naptvieJibHi npsiMi, aJCi neperU'HaK>Tb CTOpoHU 1CyTa, BiiJTU'Ha"IOTb 'Ha oiJHiU 4ozo CTopoHi piBHi BiiJpialCU, TO 80HU BiiJTU'HaK>Tb piBHi BiiJpiaxu i Ha iJpyziu
4ozo CTOpOHi (MaJI. 131).
,ZJ; o Be /1. e H HSI. Hexa:H A., A2, Aa - TO'IKH nepeTHHY napa-
MaJI. 131
8 KJiac |
|
|
|
|
|
|
|
|
|
|
|
86 |
|
|
JieJibHHX npHMHX 3 OAHi6IO ia CTOpiH KyTa |
||||||||||||
|
i A 211emHTh Mim Ai i A 3 (MaJI. 131). Hexa:H |
||||||||||||
|
Bi, |
B2, |
8 3 - |
BiAnoBiAHi TO'IKH nepeTHHY |
|||||||||
|
IJ,Hx |
npHMHX |
a |
APYl'OIO cropoHoIO |
KyTa. |
||||||||
|
,lJ;oeeAeMo, ~o |
KOJIH |
A1A2 = |
A 2A3, |
TO |
||||||||
|
B1B2 = |
B 2B3. |
|
|
|
|
|
|
|
|
|
||
|
IIpoeeAeMo qepea TO'IKYB 2 nps:My EF, |
||||||||||||
|
napa11eJibHY npHMiH A 1A 3. 3a e11acTHelcTIO |
||||||||||||
|
napa11e11orpaMa |
A1A2 = |
FB2, |
|
A~3 = |
||||||||
|
= B2E. |
OcKiJibKH |
A1A 2 = A~3, |
|
TO |
||||||||
|
FB2 = B2E. |
|
B2B 1F |
|
B2BaE |
|
|
|
|||||
|
TpHKYTHHKH |
i |
piBHi |
aa |
|||||||||
|
APYl"OIO |
oaHaKoIO. Y HHX |
B2F = |
B 2E |
aa |
||||||||
<l>anec MineTCLKHH - |
AOBeAeHHM. KyTH npH BepmHHi B2 pieHi |
||||||||||||
stK eepTHKaJibHi, a KYTH B2FB1 i B~B3 pie- |
|||||||||||||
AaBHLOrpe~LKHH yqeHHH |
|||||||||||||
ai |
$1K |
BHyTpimHi |
pi3HOCTOpOHHi |
npH |
|||||||||
(VI CT. AO H. e.) |
|||||||||||||
napa11e11bHHX npHMHX AJE3 i A1B1 i ciq- |
|||||||||||||
|
|||||||||||||
|
Hi:H |
EF. |
3 piBHOCTi TpHKYTHHKiB BHilJIH- |
||||||||||
|
Ba6 |
piBHiCTb |
cTopiH: |
B1B2 = |
B2B3. Teo- |
||||||||
|
peMy AOBeAeHo. |
|
|
|
|
|
|
|
|
||||
3 a y B a jK e H H H. B yMoei |
TeopeMH |
<1>a11eca |
aaMiCTb |
cTopiH |
|||||||||
KyTa MO)KH8 B3HTH AOBiJibHi ABi npHMi, npH J.\bOMY BHCHOBOK TeopeMH 6yAe TaKHM C8MHM:
n.apa1'eJ£bHi npsitd, R.1'i neperUH.aKJTb iJsi iJa1'i npsud i siiJTuHa- K>Tb H.a oiJ,,,iu npsMiu pis"i siiJpi31Cu, siiJruH.aKJTb pisui siiJpiaJCu i Ha iJpyzifi npR.MiU.
lHKOJIH TOOpeMa <1>a11eca BHKOpHCTOByeaTHMeTbCH i B TaKiH
cPOPMi. |
|
11' |
3 a A a q a (48). IloAiJiiTh AaHHH eiApiaoK AB Han piBHHX |
v |
'18CTHH. |
Man. 132
§ 6. qOTHPHKyTHHKH |
87 |
Po a B'.11 a a H H .11. |
IIpoBe,n;eMo a ToqKH A rriBrrp.11My a, |
~o He Jie:>KHTb Ha rrpaMiH AB (MaJI. 132). Bi,n;KJia,n;eMo Ha rriBIIpHMiH a piBHi Bi,n;pi3KH: AAi, A1A2, A2A3, ..., An-1An. CrroJiyqHMO TOqKH An i B. IIpoBe,n;eMo qepea ToqKH A1, A2, ..., An-1 rrp.11Mi, rrapaJieJibHi rrp.11Miii AnB. BoHH rrepeTHYTb Bi,n;piaoK AB y TQqKaX Bi, B2, ••• , Bn-1, ~o ,n;iJIHTb Bi,n;piaoK AB Ha n piBHHX Bi,n;piaKiB (aa TeopeMOIO <l>aJieca).
58. CEPEAJl.H JIIHUI TPHKYTHHKA
Cepeanbo10 Jtinie10 TpHKYTHHKa HaaHBaeTbc.11 Bi,n;piaoK, HKHH CIIOJJyqae cepe,n;HHH ,Zl;BOX ii:oro CTOpiH.
Teo p e Ma 6.7. Cepeihisi JiiHisi rpu-,cyrHuxa, sixa cnoJ£y11,ae cepeOUHU iJBOX iJaHUX CTOpiH, napaJ£eJ£bHa rperiu cropoHi i iJopi8HJOe ii noJ£08UHi.
,n; o Be ,n; e H H .11. Hexaii: DE - cepe,n;H.11 JiiHi.11 TPHKYTHHKa ABC (MaJI. 133). IlpoBe,n;eMo qepea TOqKY D rrp.11My, rrapaJieJihHY cTopoHi AB. 3a TeopeMoro <l>aJieca BOHa rrepeTHe Bi,n;piaoK AC B ii:oro cepe,n;HHi, T06To MiCTHTb cepe,n;Hro JiiHiro DE. 0Tme, cepe,n;aH JiiHi.11 DE rrapaJieJibHa CTOpoHi AB.
[
[
A |
B |
A |
H |
D |
MaJJ. 133 |
|
|
MaJJ. |
134 |
IlpoBe,n;eMo Terrep cepe,n;HH> JiiHiIO DF. BoHa rrapaJieJibHa cTopoHi AC. qOTHPHKYTHHK AEDF - rrapaJieJiorpaM. 3a BJiaCTHBicTro rrapaJieJiorpaMa ED= AF, a qepea Te, ~o aa TeopeMOIO <l>aJieca
1 |
|
|
AF= FB, To ED= 2 AB. TeopeMy ,n;oBe,n;eHo. |
|
|
3 a A a q a (55). ,lJ;oBe,n;iTb, ~o cepe,n;HHH CTOpiH qQTHpH- |
||
KYTHHKa e BepmHHaMH rrapaJieJiorpaMa. |
|
|
P o a B'.11 a a H H H. Hexaii: ABCD - |
,n;aHHH |
qoTHpHKYT- |
HHK i E, F, G, H - cepe,n;HHH ii:oro cTopiH (MaJI. |
134). Bi,n;pi- |
|
aoK EF - cepe,n;H.11 JiiHi.11 TPHKYTHHKa ABC. ToMy EF 11 AC. |
||
Bi,n;piaoK GH - cepe,n;H.11 JiiHi.11 TPHKYTHHKa ADC. ToMy GH II |
||
AC. 0Tme, EF II GH, T06To rrpoTHJie:>KHi |
cTopoHH EF i GH |
|
8 KJiaC |
88 |
110THpHKYTHHKa EFGH napaJieJILHi. TaK caMo ~OBO,ll;HMO napaJieJILHiCTL ,zi;pyro'i napH npOTHJiemHHX CTOpiH. 0Tme, qoTHPHKYTHHK EFGH - napaJieJiorpaM.
59. TPAil~l.H
Tpaneqie10 Ha3HBaeTLCSI 'IOTHPHKYTHHK, y S1Koro TiJILKH ~Bi npOTHJiemHi CTOpoHH napaJieJILHi. ll;i napaJieJI&Hi CTOpOHH Ha3HBftIOTLCSI OCHOBQ.MU Tpane~l. ,lJ;Bi iHmi CTOpOHH HaaHBaIOTLCSI 6i'lnuMu cTopoHaMH.
Ha MaJIIOHKY 135 6a'IHMOTpane~10 ABCD a OCHOBRMH AB i CD i 6i'IHHMHCTOpOHRMH BC i AD.
Tpane~SI, y SIKO'i6i11Hi CTOpOHH piBHi, HR3HBR6TLCSI pieno6i'l.H010. Bi,zi;piaoK, SIKHH cnoJiyqae cepe,zi;HHH 6i11HHX cTopiH, HaaHBaeTLCSI cepeanbo10 J1.inie10 Tpane~l.
[ |
0 |
B |
c |
|
|
|
or=:==1P |
|
|
|
|
~~ |
||
B |
A |
A |
0 |
E |
MaJI. 135 |
|
|
Man. 136 |
|
Teo p e M a 6.8. CefH!iJna J£inia rpane1fi'i napaJ£eJ£bHa ocnoBaM |
||||
i iJopiBn'IOe ix niscy.1td. |
|
|
|
|
.n; o Be ,zi; e H HSI. |
Hexaii: |
ABCD - |
,zi;aHa Tpane~SI (MaJI. |
136). |
IlpoBe,zi;eMo qepea BepmHHY B i cepe,zi;HHY P 6i'IH01cTopoHH CD npS1My. BoHa nepeTHHae npSIMy AD y ,zi;ea:Ki:H TO'I~ E.
TpHKYTHHKH PBC i PED piBHi aa ,zi;pyro10 oaHaKoIO piBHOCTi TPHKYTHHKiB. Y HHX CP =DP aa no6y,zi;oBoIO, KYTH npH BepmHHi P piBHi SIK BepTHKRJibHi, a KYTH PCB i PDE piBHi SIK BHYTpimHi piaHOCTOpoHHi npH napaJieJI&HHX npSIMHX BC i AD i ci11Hi:H CD. 3 piBHOcTi TpHKYTHHKiB BHnJIHBae piBHicT1> cTopiH PB= PE, BC= ED.
Orme, cepe,o;HR JiiHiR PQ Tpane~'i e cepe,zi;HhOIO JiiHi610 TPHKYTHHKa ABE. Sa BJiaCTHBicTIO cepep;Hi.o'iJiiHil TPHKYTHHKa PQ
i eip;piaoK PQ = {-AE = ~AD + BC). TeopeMy ,zi;oBe,zi;eHo.
3 a p; a q a (6~)..,ll;o.ee,zi;iTL, w;o y |
piBHo6i•rni:H |
Tpane~fi |
KYTH npH OCHOBl plBHl. |
|
|
Po a B'SI a a H HSI. Hexa:H ABCD - |
piBHo6i11Ha |
Tpane~SI |
(MaJI. 137). ,lJ;oBep;eMo, w;o KYTH Tpane~'i npH ocHoBi CD piBHi.
IlpoBep;eMo qepea BepWHHY B npS1My, napaJieJILHY cTopoHi
§ 6. 'IOTHpHKYTHHKH |
|
|
|
|
|
89 |
||
AD. BoHa nepeTHe npoMiHb |
DC |
A |
|
B |
||||
y Ae~rniii TO'i~ E. qOTHPHKYTHHK |
|
|
1 |
|||||
i•-----D- |
||||||||
ABED - |
napaJieJiorpaM. 3a BJiac- |
|||||||
THBiCTIO |
napaJieJiorpaMa BE = |
|
|
|
||||
= AD. |
3a |
YMOBOIO |
AD = BC |
|
E |
[ |
||
(Tpane:u;i.a piBH06i'IHa),OT)Ke, TpH- |
0 |
|||||||
KYTHHK |
BCE |
piBH06eApeHHH |
a |
|
Man. 137 |
|
||
OCHOBOIO |
EC. |
KyTH |
TpHKYTHHKa |
|
|
|||
|
|
|
||||||
i Tpane:u;ii npH BepmHHi C a6iraIOTbC.a, a KYTH npH BepmHHax E i D piBHi .aK Bi)uIOBiAHi KYTH npH nepeTHHi napaJieJibHHX np.aMHX ci'IHOIO. ToMy L ADC= L BCD. TBepAmeHH.a AOBeAeHO.
60. TEOPEMA IIPO IIPOIIOP~IRHI BIAJ>I3KH
Te ope M a 6.9. llaptUeJ&'bHi npsuai, J&Xi neperuHalOT'b crop01iu xyra,, 6WTUH410T'b si8 CTOpiH xyra, nponOPlfiUHi 6Wpiaxu.
,n; o B e A e H H a. |
Hexaii cropoHH KyTa A |
nepeTHHaIOTb napa- |
|
JieJibHi np.aMi - y |
TO'iKaXB, C i Bi. Ci BiAnOBiAHO (MaJI. 138). |
||
TeopeMa CTBepAmys, m;o |
|
|
|
|
AC 1 |
AB1 |
(•) |
|
AC= AB. |
||
,IJ;oBeAeMO CilO'iaTKYpiBHicTb (*) AJIR BHilaAKY, KOJIH iCHYG BiApiaoK TaKOl AOB)KHHH 6, .aKHH MO)KHa Bi.AKJiaCTH ~Jle 'iHCJIOpaaiB i Ha Bi.ApiaKy AC i Ha Bi.ApiaKy AC1. Hexaii AC= n6, AC1 = m6 i n > m. Pooi6'GMoBi.ApiaoK AC Ha n piBHHX qacTHH (AOBmHHOIO 6). IlpH D;bOMY TO'iKacI 6yAe OAHiGIO 3 TO'iOKnoAiJiy. IlpoBeAeMO qepea TO'iKHnoAiJIY np.aMi, napaJieJibHi np.aMiA BC. 3a TeopeMoIO <l>aJieca ~ np.aMi poo6HB&IOTb Bi.ApiaoK AB Ha piBHi BiApiaKH Ae&Ko'iAOB)KHHH 61. MaeMo:
Ba'iHMo,m;o
AC1 m . |
AB1 m |
AC= n 1 |
AB= n |
Man. 138 |
Man. 139 |