Luyện thi Toán học - Chuyên đề trọng điểm bồi dưỡng học sinh giỏi Hình học không gian: Phần 1

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Phần 1 tài liệu Chuyên đề trọng điểm bồi dưỡng học sinh giỏi Hình học không gian do Nguyễn Quang Sơn biên soạn cung cấp cho người đọc các kiến thức: Đại cương hình học không gian, quan hệ song song trong không gian, quan hệ vuông góc. Mời các bạn cùng tham khảo.

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Nội dung Text: Luyện thi Toán học - Chuyên đề trọng điểm bồi dưỡng học sinh giỏi Hình học không gian: Phần 1

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- Cho duong thang d va diem A T i n h c h a t 2: C o mot v a chi m o t m a t .A khong thuoc d. K h i do diem A va p h a n g di q u a ba d i e m idiong thang hang. duong thSng d xac dinh mpt mat p h S n g , k i h i e u la m p ( A , d ) , h o a c m p T i n h c h a t 3: Ne'u m o t d u o n g t h a n g c 6 (d, A ) h a y ( d . A ) . hai diem phan bi^t thuoc mpt mat M a t p h S n g d u p e hoan toan xac djnh khi biet no c h i i a hai d u o n g thSng c3t nhau: p h a n g thi m p i d i e m c u a d u o n g t h S n g C h o h a i d u o n g t h a n g cat n h a u a v a b . d e u thuQC m a t p h a n g do. • D K h i d o h a i d u o n g t h a n g a v a b xac d j n h m p t m a t p h a n g v a ki h i e u la m p ( a , b) T i n h c h a t 4: T o n tai b o n d i e m k h o n g cung thuoc mpt mat ph5ng . h a y (a , b ) , h o a c m p (b , a) hay (b, a ) . T i n h c h a t 5: N e u hai m a t p h a n g p h a n bit^t C O m p t d i e m c h u n g thi c h u n g c o n CO m p t d i e m c h u n g k h a c n u a . T u tinh c h a t n a y s u y ra: Ne'u hai m a t H I N H CHOP VA Tir DIEN ,!'; ' p h a n g p h a n biet c 6 m p t d i e m c h u n g thi 1. Khai niem: c h u n g se c 6 mot d u u n g t h a n g c h u n g di T r o n g m a t p h ^ n g ( a ) c h o d a giac loi A , A 2 A 3 . . . A ^ . L a y m p t d i e m S k h o n g qua diem chung ay. Duong thang c h u n g la d u y nhiit c h u a tat ca cac d i e m t h u o c m a t p h ^ n g ( a ) . L a n l u p t noi d i e m S v o i cac d i n h A ^ , A j , A 3 , . . . , A,., ta c h u n g cua hai mat phang do . D u o n g dupe n tam giac S A j A j , SAjA-,,..., SA,^A,. Hinh gom da giac t h a n g c h u n g d o d u p e goi la giao t u y e n AjA2A3...A„ va n t a m giac S A j A j , SA2A3 S A ^ A , d u p e gpi la hinh ciia hai m a t p h a n g . chop, ki h i e u la S . A , A 2 A 3 . . . A , , . T a gpi S la dinh cm hinh chop, c o n d a giac V i d u : D u o n g t h a n g c h u n g d c u a hai m a t p h a n g p h a n biet ( a ) v a ( p ) d u p e A , A 2 A 3 . . . A | , la mat day cm hinh chop, cac t a m g i a c SAjAj, SA2A3, gpi la G I A O T U Y E N c u a hai mat p h a n g ( a ) v a v a ki hieu la d = ( a ) n ( p ) . SAj^Ajdupc gpi la cac mat hen cm hinh chop, cac doan thang T i n h c h a t 6: 7 ren m o i m a t p h a n g , cac ket q u a d a biet trong h i n h hpc p h a n g deu dung. S A , , S A 2 , S A 3 , . . . , S A , , d u p e gpi la cac canh ben ciia hinh chop . III. C A C H X A C D I N H M O T M A T PHANG T a gpi h i n h c h o p c 6 d a y la tam giac, t u giac, ngij giac,..., l a n l u p t la hinh chop C o ba each x a c d j n h m p t m a t p h a n g : tam i^idc, hinh chop ti'f giac, hinh chop ngu giiic, .... - Mat phang dupe hoan toan xac C h o b o n d i e m A , B , C , D k h o n g d o n g p h S n g . H i n h g o m b o n t a m giac d j n h k h i biet n o d i q u a ba d i e m k h o n g A B C , A C D , A B D v a B C D gpi la hinh tie dien (hay ngMn g p n gpi la tu dien) va thang hang. ' :j d u p e k i h i ^ u la A B C D . C a c d i e m A , B, C , D gpi la c a c dinh cm tie dicn. Cac - M a t p h a n g d u p e h o a n toan xac d j n h d o a n t h S n g A B , B C , C D , D A , C A , B D gpi la c a c canh cm tu dien . H a i canh khi biet n o d i q u a m p t d i e m v a c h u a k h o n g d i q u a m p t d i n h gpi la hai canh doi dien cua tie dien. C a c t a m giac A B C , mpt d u o n g thang k h o n g di q u a d i e m do. A C D , A B D , B C D gpi la cac mat ciia tie dien . D i n h k h o n g n S m tren m a t gpi la dinh doi dien cm mat do . ,;V... •• %' 4
H i n h tu d i # n c6 b o n m 3 t la cdc tarn gidc deu g p i la h i n h tii diftt deu . BAI TAP G I A I CHI T I E T A S Co b o n d g n g toan chi'nh la: T i m giao t u y e n cua hai m a t p h 3 n g . T i m giao d i e m ciia d u o n g th3ng va mat phang. T i m thiet di?n ciia h i n h chop. C h u n g m i n h ba d i e m thJing hang, ba d u o n g thMng d o n g q u i ... Ta Ian l u p t xet t u n g d ^ n g m p t n h u sau: D A N G 1: Tim giao tuyen cua hai m|t phang. ,1. , Phuong phap: M u o n tim giao tuyen ciia hai mat phMng? Ta t i m hai d i e m chung thuQc ca hai mat p h a n g . N61 hai d i e m chung do du
fOe AC,ACc:(SAC) , ^ , . b) T i m giao t u y e n cua m p (IBC) va m p ( D M N ) . V\\ /=^Oe{SAC)n(SBD) (2) OeBD,BDcz(SBD) V / V ; v^ T r o n g m p ( A B D ) goi E = BI n D M , v i : E€BI,BIc(lBC) Tir (1) va (2) suy ra (SAC) n ( S B D ) = S O . EeDM,DMc(DMN) b) Cau b c u n g t u o n g t u (SAB) va (SCD) c6 d i e m c h u n g t h i i nhat la S, d i e m c h u n g t h i i hai la E, v o i E la giao d i e m ciia A B va C D v i hai d u a n g thSng nay =>Ec:(lBC)n(DMN) (l) c u n g thuQC m a t p h a n g ( A B C D ) va chung k h o n g song song v o i n h a u . Cach T r o n g m p ( A C D ) goi F = C I n D N , v i : g t r i n h bay c u n g n h u cau a): FeCI,CIc(lBC) Ta CO S € ( S A B ) n ( S C D ) (3) '''' • > 'FGDN,DNC(DMN) •'" •^- • • ' i ' ; T r o n g m p ( A B C D ) goi E = A B n C D , v i : ' ^''^ =^Fc(lBC)n(DMN) (2) , jEe AB,ABc(SAB) Ee(SAB)n(SCD) (4) Tu'(l)va (1): ( i B C ) n ( D M N ) = E F 5 |EGCD,CDC{SCD) Cho t u d i e n A B C D . Lay cac d i e m M thuoc canh A B , N thuoc canh A C sao Tir (3) va (4) suy ra ( S A B ) n ( S C D ) = SE cho M N cat BC. Goi I la d i e m ben trong tam giac BCD. T i m giao tuyen ciia: c) Cau nay c u n g vay S la d i e m chung t h u nhat, d i e m c h u n g thiV hai la giao a) M p ( M N I ) va m p (BCD). d i e m ciia A D va BC, v i hai d u o n g thSng nay cung thuoc m p ( A B C D ) va b) M p ( M N I ) va m p ( A B D ) . c h i i n g k h o n g song song, ta t r i n h bay n h u sau: c) M p ( M N I ) va m p ( A C D ) . Taco SG(SAD)n(SBC) (5) LOI GIAI ( A B , C D e ( A B C D ) ) Goi F A D n BC, v i : a) M p ( M N I ) va m p (BCD). FeAD,ADc(SAD) Goi H = M N n B C ( M N , B C c ( A B C ) ) Fe(SAD)n(SBC) (6) FeBC,BCc(SBC) Taco: I e ( l M N ) n ( B C D ) (l) // Tir (5) (6) suy ra ( S A D ) n (SBC) = S F . HeMN,MNc(lMN) va < Cho t u dien A B C D . Goi I , J Ian l u g t la t r u n g d i e m cac canh A D , BC. HeBC,BCc(BCD) a) T i m giao t u y e n ciia 2 m p (IBC) va m p (JAD). , < =^He(lMN)n(BCD) (2) b) Lay d i e m M thuoc canh A B , N thuoc canh A C sao cho M , N k h o n g la t r u n g d i e m . T i m giao tuyen ciia m p (IBC) va m p ( D M N ) . Tu(1)va(2): (IMN)n(BCD) =HI LOI GIAI b) M p ( M N I ) va m p ( A B D ) . a) T i m giao t u y e n ciia 2 m p (IBC) va m p (]AD). T r o n g m p (BCD) goi E va F Ian l u g t la giao d i e m cua H I v o i B D va C D . I e (IBC) M e (MNI) Co , :^Ie(lBC)nfjAD) (1) '..fin th fM Co r>Me(MNl)n(ABD) (3) i^'^^''' I e A D , A D c (JAD) / ' ' Me ABc(ABD) ••; Je(jAD).,, EeHIc(MNl) Va (2) ^ Va EG(MNl)n(ABD) (4)' j6BC,BCc(lBC) ^ ' ^ ' EGBD(=(ABD)~ T u (1) va (2) ^ ( I B C ) n ( J A D ) = IJ T u (3) va (4) suy ra ( M N I ) n ( A B D ) = M E . ^,
c) M p ( M N I ) va m p ( A C D ) . Cho h i n h chop S.ABCD day la h i n h binh hanh t a m O. G p i M , N , P Ian l u p t la Ne(MNl) t r u n g d i e m cac canh BC, C D , SA. T i m giao tuyen cua : Co N e ( M N I ) n ( A C D ) (5). ''" • ' ^ i N e AC c (ACD) a) M p ( M N P ) va m p (SAB). b) M p ( M N P ) va m p (SAD). ,,•4. c) M p ( M N P ) va m p (SBC). • d ) M p ( M N P ) va m ^ (SCD). FeHIc(MNl) , . , V a \) ( 6 ) . LOI GIAI S FeCDc(ACD) , V, „ ^ /, , Gpi F = M N n A B , E = M N n A D T u (5) va (6) suy ra ( M N I ) n ( A C D ) = N F (vi M N , AB, A D c (ABCD)) Cho h i n h chop S.ABCD c6 day A B C D la h i n h thang c6 A B song song C D . a) M p ( M N P ) va m p ( S A B ) . T i d v Gpi I la giao d i e m cua A D va BC. Lay M thupc c^nh SC. T i m giao tuyen cua: P6(MNP) Co a) M p (SAC) va m p (SBD). • iymf^rxlOm}^ -i P£SA,SAC(SAB) b) M p (SAD) va m p (SBC). P€(MNP)n(SAB) (l) c) M p ( A D M ) va m p (SBC). LOIGIAI FeMN,MNc(MNP) Co a) T i m giao t u y e n cua m a t phcing (SAC) va (SBD). F€AB,ABc(SAB) Taco S € ( S A C ) n ( S B D ) (l) =>Fe(MNP)n(SAB) (2) ^ T r o n g m p ( A B C D ) g p i H = A C n B D , c6: T u (1) va (2) suy ra ( M N P ) n ( S A B ) = P F ' J H e A C c (SAC) b) M p ( M N P ) va m p (SAD). ^HeBDc(SBD) PG(MNP) =>H€(SAC)n(SBD) (2) ^ Ta c6: • =:>PG(MNP)n(SAD) (3) PeSA,SAc(SAD) T u ( l ) v a (2) suy ra ( S A C ) n ( S B D ) = SH . EeMN,MNc(MNP) b) T i m giao t u y e n ciia m a t ph^ng (SAD) va (SBC). Va CO Ee(MNP)n(SAD) (4) Ee A D , A D c ( S A D ) Taco: S € ( S A D ) n ( S B C ) (3) Tir (3) va (4) suy ra ( M N P ) n ( S A D ) = PE . it': T r o n g m p ( A B C D ) gpi I = A D n B C , c6: I e A D c (SAD) c) M p ( M N P ) va m p (SBC). UlE(SAD)n(SBC) (4) T r o n g mp(SAB) g p i K = P F n SB , c6: I € BC c (SBC) fKGPF,PFc(MNP) ' Id ^-^^^9 Tir (1) va (2) suy ra ( S A D ) n (SBC) = S I . ,1 H i, /: 'K K €eSSB,SB B , S B ec ( S BC) (SBC) " ^ ^^^^^ c) T i m giao t u y e n m p ( A D M ) va m p (SBC). 1 Ui [ M e (MNP) [ M € ( A D M ) ^ ' ^ =>Me(ADM)n(SBC) Vac6M6(MNP)n(SBC) (6) MeSC,SCe(SBC) ^ > y f (5) MeBCBCc(SBC) ^ / v / jle AD,ADc(ADM) Tir (5) va (6) suy ra ( M N P ) n (SBC) - M K •l€(ADM)n(SBC) (6) IeBC,BCc(SBC) d) M p ( M N P ) va m p (SCD). Tir (5) va (6) suy ra ( A D M ) n ( S B C ) = M I -; {I) U7 i;T G p i H - P E n SD ( P E , SD c ( S A D ) ) , c6:
a). Tim giao tuyen cua (AMN) va (BCD) HGPE,PECI(MNP) He{MNP)n(SCD) (7) Ee A M , A M c ( A M N ) HGSD,SDCI(SCD) Trong (ABD ) goi E = A M n BD, c6: EGBD,BDC(BCD) Ne(MNP) Va CO ^ / Ne(MNP)n(SCD) (8) \ =^ E G ( A M N ) n (BCD) (l) , , [ N € C D , C D C ( S C D ) Trong (ACD ) gpi F = A N n CD, c6: Tir va (8) suy ra: (MNP) n (SCD) = NH FG A N , A N C ( A M N ) Cho t u dien S.ABC. Lay M e S B , N e AC, l e S C sao ciio M I khong song ' F G C D , C D C ( B C D ) song vai BC, N I khong song song voi SA. Tim giao tuyen ciia mat phang (MNI) voi cac mat (ABC) va (SAB). ^FG(AMN)n(BCD) (2) LOI GIAI Tu(1) v a ( 2 ) : ( A M N ) n ( B C D ) = F.F ^ a) Tim giao tuyen cua mat phang ( M N I ) va (ABC). b. Tim giao tuyen cua ( D M N ) va (ABC) NG(MNI) ^ Trong (ABD ) goi P = D M n AB , c6: Yi \ P G D M , D M C ( D M N ) N G AC,ACc(ABC) • PG(DMN)n(ABC) (3) P G AB,ABc(ABC) =>NG(MNl)n(ABC) (l) Trong (ACD) goi Q = D N n AC , c6: Trong mp(SBC) goi K = M I n BC, vi: Q G D N , D N C ( D M N ) •ml KGMIC=(MNI) ^ ' : ^ Q e D M N n(ABC) (4) Q G AC,ACc(ABC) ^ ' ^ ' ^ ' K G B C , B C C ( A B C ) =:>Ke(MNl)n(ABC) Tu(3)va(4) : ( D M N ) r , ( A H C ) - I'Q T u (1) va (2) suy ra: Cho tii dien ABCD. Lay I e AB, ] la diem trong tam giac BCD, K la diem (MNl)n(ABC) = N K trong tam giac ACD. Tim giao tuyen cua mat phang (IJK) voi cac mat b) Tim giao tuyen cua mat phang (MNI) voi (SAB). cua t u dien / LOI GIAI Goi J= NI n SA(NI, SA c (SAC)) Goi M - DK n A C ( D K , AC c ( A C D ) ) ; N = DJ n B C ( D ] , BC cz (BCD)) Me (MNI) Ta co: >M6(MNl)n(SAB) (3) Va: H = M N n K j ( M N , K J c ( D M N ) ) . MGSB,SBC(SAB) Vi H G M N , M N c (ABC) :=> l i G ( A B C ) 'JGNIC(MNI) Va CO J G ( M N I ) n (SAB) (4) JeSA,SAc(SAB)' Goi P - H I n B C ( H I , BC c (ABC)), Tu (3) va (4) suy ra: (MNI) n (SAB) = MJ. Q - PJr,CD(PJ,CDc(BCD)) Cho tu dien ABCD, M la mot diem ben trong tam giac ABD, N la mot T = QK n AD(QK, AD c (ACD)) diem ben trong tam giac ACD. Tim giao tuyen cua cac cap mp sau : Theo each dung diem a tren ta c(S: a) ( A M N ) va (BCD) " M'Ti;^f7 • '- ( I J K ) n ( A B C ) = IP; (IJK)n(BCD) = PQ b) (DMN) va (ABC) ( l J K ) n ( A C D ) = Q T ; ( l J K ) n ( A B ) = TI Wi • M"/ LOI GIAI 13
D A N G 2: Tim giao diem ciia duong thang va m|t phing. Cho tu dien A B C D . Tren A B , A C , B D Ian luot lay 3 diem M , N , P sao cho Phuang phap: Muon tim giao diem cua duong thiing d va m|t phling (P), M N khong song song voi B C , M P khong song song voi A D . Xac djnh giao CO hai each lam n h u sau: diem cua cac duong thing B C , A D , C D voi mp (MNP). Cach 1: N h u n g bai don gian, c6 san mot mat phaiTg (Q) chua duong thSng d LOIGIAI , va m p t duong thSng a thuQC mat ph3ng (P). > s Tim giao diem cm BC va mp(MNP) Giao diem ciia hai duong thang khong Gpi H = M N n B C ( M N , B C c ( A B C ) ) . song song d va a chinh la giao diem ciia d va mgt phang (P). HeBC Ta c6: Cach 2: T i m myt mat phang (Q) H e M N , M N c: ( M N P ) chua duong thang d, sao cho de r:>H = B C n ( M N P ) dang tim giao tuye'n voi mat phang (P). Giao diem ciia duong thang d Tim j^iao diem cua AD va mp(MNP) ^ va mat phang (P) chinh la giao diem G
b) . Tim giao diem cua BC voi (OMN): Cho t u dien ABCD. Gpi M , N Ian iupt la trung diem cac canh AC, BC. Gpi P = B C n O l ( B C , O I c : ( B C D ) ) . Tren canh BD lay diem P sao cho BP = 2PD. Lay Q thupc AB sao cho Q M Vay : P = BC o ( O M N ) cat BC. Tim: c) . Tim giao diem cua BD voi (OMN); a) . Giao diem ciia CD va mp (MNP). Trong (BCD), goi Q = BD n OI. b) . Giao diem ciia A D va mp (MNP). Vay : Q = BD n (OMN). c) . Giao tuyen ciia mp (MPQ) va mp (BCD). d) . Giao diem ciia CD va mp (MPQ). Cho t u di^n ABCD, lay M thupc AB, N thupc AC, I la diem thupc mien trong d). Giao diem ciia A D va mp (MPQ). cua tam giac BCD. Xac dinh giao diem cua cac duong thMng BC, BD, AD, CD voi mp (IMN). LOI GIAI a) . Giao diem cua CD va mp (MNP). /in/- , LOl GIAl Tim giao diem ciia BC va mp(MNI) Gpi E = C D n NP(CD,NP c mp(BCD)). < HeBC ' •if"-' frnT .! EeCD Gpi H = MNOBC(MN,BCC(ABC)),C6: Co: , ^ =>E = C D n ( M N ] V H€MN,MNC(MNI) [EeNP,NPc(MNP) ^ =*H = BCn(MNI) \ b) . Giao diem ciia A D va mp (MNP). Tim giao tuyeit cua (BCD) lui (MNI) viT: Tim giao tuyen ciia m p ( A C D ) va mp(MNP). HGMN, MNC(MNI) M€(MNP) la c6: { ' ' He(MNl)n(ACD) Co [HeBC, BCc(BCD) Me AC,ACC(ACD) K Va l e ( M N l ) n ( A C D ) (2), Me(MNP)n(ACD) (l) Tir (1) \i (2): H1 = (M NI) n (ACD) EeNP,NPc{MNP) Co Tim giao diem ctia BD va mp(MNl) EeCD,CDc(ACD) Co\ = HI o DD ( H I , BD c (BCD)), c6: =>E6(MNP)n(ACD) (2) FeBD ^ " Tu (1) va (2): EM = ( M N P ) n (ACD) • = nDn(MNl) 1-6Hl,Hic(MNl) Gpi F - A D n E M ( A D , E M c ( A C D ) ) ^ F=ADn(MNP)(vi E M C ( M N P ) ) Tim giao diem ciia CD va mp(MNI) c) . Tim giao tuyen ciia mp (MPQ) va mp (BCD). , Gpi F = HInCD(Hl,CDc(BCD)),c6: Gpi K = Q M n B C ( Q M , B C c m p ( A B C ) ) 1 - iYa}?\'m^J^f^M^^^ FGCD F-CDn(MNl) , (KeBC,BCc(BCD) FGH1,H!C(MNI) Co ^Ke(MPQ)n(BCD) (3) KGQM,QMC(MPQ) Tim giao diem cua AD vii mpiMNl) Pe (MNQ) Co Pe(MPQ)n(BCD) (4) Gpi K = F N n A D ( F N , A D c ( A C D ) ) , c6: PeBD,BDc(BCD) ~ =>K = A D n ( M N l ) Tu (3) va (4): KP = ( M P Q ) n (BCD). d). Tim giao diem c i i a CD v a m i i ^ T ^ V J H N T i M H O I N f i Tl lUAN
Gpi L = K P O C D ( K P , C D C ( B C D ) ) , C 6 : d). Tim giao diem P ciia SC va mp (ABM), suy ra giao tuyen ciia hai mp (SCD) LeCD , ^ va (ABM). Tim giao tuyen ciia hai mat phang (AMB) va (SAC). , Ta CO A va I la hai diem chung ciia hai mat phSng (ABM) va (SAC). e). Tim giao diem ciia A D va mp ( M P Q ) . ifc *.J = M N n ( S B D ) JGSQC(SBD) ^ ' c) Tim giao diem I ciia duong thing BM va mat ph5ng (SAC). M l s ' l ! Cho hinh chop S.ABCD .Gpi O la giao diem ciia AC va BD . M , N , ? Ian lupt Gpi I = B M n S O ( B M , S O e (SBN)). 5.'11 ^ ' ' la cac diem tren SA , SB , SD. a) . Tim giao diem I ciia SO voi mat phang ( MNP ) Co: . I = BMn(SAC). l£SO,SOc(SAC) ^ b) . Tim giao diem Q ciia SC voi mat ph^ng ( MNP ) ^ ____ 18 19
' LOIGIAI a) Tim giao diem I ciia SO voi mat pinang ( MNP ) b) Tim giao diem ciia A O va (BMN ): ! '•[. \ ' ' " f Trong mp(SBD) goi : I = SOr>NP, co: ^ H,iti • ChQn mp (ABP) 3 AO. "leSO , , '-.'S''" Tim giao tuyen cua (ABP ) va (BMMj l€NPc(MNP) ^ ' Ta c6:B la diem chung cua (ABP ) va (BMN) (3) QGMN,MNC:(BMN) , , , . , , b). Tim giao diem Q ciia SC voi mp (MNP ) / , ' ^ Q G ( A B P ) n ( B M N ) (4). • Chon mp phu (SAC) 3 SC ^ Q G A P , APc(ABP) ^ ' ^ ' ^ ' • Tim giao tuyen ciia ( SAC ) va (MNP) Tir (3) va (4) ( A B P ) n ( B M N ) = BQ ' f ^v?) D „\i::, M € (MNP) Ta c6: Goi I = BQ n A O ( vi BQ, AO c mp( ABP)), c6: MeSA,SAc:(SAC) leAO , , r\(m\ MG(MNP)n(SAC) (l) IeBQ,BQc(BMN) ^ ' l6SP,SPc(MNP) Va I G (MNP) n (SAC) Trong mp (a) cho hinh thang ABCD, day Ion A D . Goi I, J, K Ian lugt la cac I e SO, SO c (SAC) diem tren SA, AB, BC ( K khong la trung diem BC). Tim giao diem ciia: T u ( l ) v a (2) CO ( M N P ) n ( S A C ) = M I a) IK va (SBD). b) SD va (IJK ). QeSC • Trong (SAC) goi Q = SC n M I , c6: c) SCva (IJK). QeMI,MIc(MNP) LOI GIAI ' ^ Q - SCn(MNP) a). Tim giao diem cua IK va (SBD) Cho t u dien ABCD. Goi M , N la hai diem tren AC va A D . O la diem ben • Chon mp phu (SAK) chua IK . trong tam giac BCD. Tim giao diem cua : Tim giao tuyen ciia (SAK ) va (SBD) a) . M N v a ( A B O ) Taco: SG(SAK)r^,(SBD) (l) b) . AO va (BMN ) Trong (ABCD) goi P = AK n BD. LOI GIAI a) Tim giao diem ciia M N va (ABO ): Pe A K c ( S A K ) Co • Chon mp phu (ACD) =) M N . Tim giao tuyen ciia (ACD ) va (ABO) P G BD c (SBD) Ta c6: A la diem chung cua (ACD ) va (ABO) =>PG(SAK)n(SBD) (2) Trong (BCD) goi P = BO n DC. Tu (I) va (2) :=> (SAK) n (SBD) = SP PeBO,BOc(ABO) PeCD,CDc(ACD) / • Trong (SAK), goi Q = IK n SP, c6: Q = IKn(SBD) QGSPC(SBD) =^ P G (ABO) n (ACD) (2) t b). Tim giao diem cua SD va (IJK ) : Tir (1) va (2) ^ (ACD) n (ABO) = AP • Chon mat phang phu (SBD) chua SD. Tim giao tuyen ciia (SBD ) va (IJK). - Trong (ACD), goi Q = AP n M N , c6: Theo cau a) ta c6: Q e (SBD) r^ ( I J K ) (3) QeMN MNn(ABO) = Q Trong (ABCD), goi M = JK n BD M la diem chung cua ( I J K ) va (SBD) (4) Q G A P , A P c ABO Tu(3)va(4)=v (IJK) n (SBD) = Q M . • ' , 20
• Trong (SBD) gQi N = Q M m SD. T r o n g m a t phSng day ( A B C ) ke A N // E F ( N e B C ) . ., r> < BH BF , „^ i/VA ' ;.. - T a c o : H F / / A N = : > : ^ = : £ ^ = 1 ^ BF = F N . HA FN c). T i m giao d i e m ciia SC va ( I J K ) : '' "'* ' T a c o : E F / / A N => — = — = 2 CN = 2NF. • C h p n m p p h u ( S A C ) Z3 S C . T i m giao tuyen cua ( S A C ) va ( I J K ) . !, AE NF FB FB FB 1 i/iae)zvCc Ket luan: Col ' ^ , ^ l 6 IJK n SAC (5) FC FN + NC 3FB 3 , [l€SA,SAc:(SAC) ^ / / ^ , . =, . I 1 uT b). T i m giao d i e m ciia K M va m p (ABC). »• •< f » • (• f . C(?i E = A C n J K ( v i A C J K c ( A B C D ) ) . Vay E e (IJK) n (SAC) (6) > , T a c o K M C ( I H K ) . Goi J = K M n E H ( E H , K M c ( I H K ) ) . T u (5) va (6) (IJK) n (SAC) = IE >; 0/.="!' • T r o n g (SAC), gpi F = I E n S C . JeKM , , Cho t u d i ^ n S.ABC. Goi I , H Ian l u p t la t r u n g d i e m ciia SA, A B . Tren canh SC lay d i e m K sao cho CK = 3 S K . Thie't d i ^ n la phan c h u n g ciia mat phang (P) va h i n h ( H ) . P A N G 3: T i m thiet dign ciia hinh (H) k h i cit bai mat phang (P). ^ ' ' Xac dinh thiei di^n la xac d i n h giao tuyen ciia m p (P) v o i cac mat ciia h i n h (H). FB a). T i m giao d i e m F cua BC v o i m p ( I H K ) . T i n h t i so T h u o n g ta t i m giao t u y e n dau t i e n ciia mat phSng (P) v a i m p t m a t phang FC b). G Q I M la t r u n g d i e m cua doan I H . T i m giao d i e m cua K M va m p ( A B C ) . ( a ) nao d o thupc h i n h ( H ) , giao tuyen nay de t i m duoc. Sau do keo dai LOI GIAI giao t u y e n nay c l t cac canh khac ciia h i n h ( H ) , t u do ta t i m d u p e cac giao a). T i m giao d i e m ciia BC v o i m p ( I H K ) . t u y e n tiep theo. Da giac gioi han boi cac doan giao t u y e n nay khep k i n Ta t i m giao tu)'e'n ciia (ABC) va ( I H K ) trudc thanh m p t thiet dien can t i m . Gpi E = A C n K l ( A C , K I c ( S A C ) ) , c 6 : T h o n g qua cvi the n h u n g bai tap sau thi cac b^n se hieu ro hom. [E€AC,ACC(ABC) Cho h i n h chop S.ABCD. G p i M la m p t d i e m t r o n g t a m giac SCD. a) . T i m giao t u y e n ciia hai m p (SBM) va (SAC). * '-^'f' b) . T i m giao d i e m ciia d u o n g thSng B M va m p (SAC). ""^'^^^ rr'i' fH6(lHK) c). Xac d j n h thie't dien ciia h i n h chop k h i cat b a i m p ( A B M ) . T u (1) va (2) => E H = ( A B C ) n ( I H K ) LOI GIAI a). T i m giao t u y e n ciia hai m p (SBM) va ( S A C ) . ' 5 V ' ' ' * Gpi F = E H n B C ( E H , BC c ( A B C ) ) , c6: Gpi N = SM n C D ( vi S M , C D c (SCD)). F e BC V^y mat phSng (SBN) cung la mat phJing (SBM). FeHH,EHe(lHK)^^ = ^C"M; Gpi 0 = ACnBN(vi AC,BNC(ABCD)). ^ Ggi D t r u n g d i e m ciia S C , ta c6 I K la d u o n g t r u n g b i n h ciia A S A D , OeBN,BNc(SBN) T r o n g ACEK c6: — = = 2 = ^ C A = 2CK . Ta c6: AE DK OG AC,ACC(SAC)
=^Oe(SAC)r^{SBN) (l) Trong mp (BCD), goi I = BD n H O . VaSe(SAC)n(SBN) (2) ^^^^^ JleBD I = BDr^(MNO ^°'|leHO,HOc(MNO) T u (1) va (2): ( S A C ) n ( S B N ) = S O . ''' Trong mp(BCD) goi ] = C D n H O , c6: b) . T i m giao diem cua duong thSng B M va mp (SAC). ' '' ' •jeCD I Gpi H = B M n S O ( v i B M , S O c ( S B N ) ) , c 6 : [J € H O , H O c ( M N O ) H H€BM r:^J = C D n ( M N O ) . •H . S0.SO c ( S A C ) " " = ""^ " '•' '"^ c). Tim thiet dien cua mp(OMN) voi hinh ch c) . Xac djnh thiet dien cua hinh chop khi cat boi mp (ABM). Theo each dung diem o cau a) va b) thl : Goi I = A H n S C ( v i A H , S C c: ( S A C ) ) . (ABC)n(MNO) = MN, (ABD)n(MNO) Tim giao tuyen cm hai mat phang (ABM) va (SCD). ( A C D ) r ^ ( M N O ) = N J , ( B C D ) n ( M N O ) = IJ. 16A H , A H c (ABM) , ' , , , , . f r. Vay thiet dien can tim la tu giac MNJI. / , ''^16 SCD n ABM) (s) l€SCSCc(SCD) \ \ \ Cho tu dien S . A B C . Goi M € A S , N e ( S B C ) , P e ( A B C ) , khong c6 duong thang nao song song. Va M e ( S C D ) n ( A B M ) (4) 'UUbvX ;| a) Tim giao diem cua M N voi (ABC), suy ra giao tuyen cua ( M N P ) va (ABC). T u (3) va (4): ( S C D ) n ( A B M ) = IM. b) T i m giao diem cua A B vol ( M N I ' ) . Goi J = I M n S D ( v i I M , S D c ( S C D ) ) . , c) T i m giao diem cua NI^ voi (SAB). d) T i m thiet dien ciia hinh chop cat boi mat phang ( M N P ) . Vay ( S A D ) n ( A B M ) = AJ , va ( S B C ) n ( A B M ) = BI t«V LOI GIAI Ket luan : thiet dien can tim la tu giac ABIJ . a). T i m giao diem cua M N voi (ABC) 5 Cho tu di^n A B C D . Tren AB, A C lay 2 diem M, N sao cho M N khong Chpn mat phang phu (SAH) chua M N . / song song B C . Goi O la 1 diem trong tarn giac B C D . Tim giao tuyen cua a) . Tim giao tuyen (OMN) va (BCD). mp(SAH) va mp(ABC). Ae(ABC)n(SAH) (l) b) . Tim giao diem cua D C , B D voi (OMN). c) . Tim thiet dien ciia mp(OMN) voi hinh chop. Trong mp(SBC), goi: / LOI G I A I H = SN n BC . a) . Tim giao tuyen ( O M N ) va (BCD). HeSN,SNc(SAH) Co: [HeMN,MNc(MNO) HeBC,BCe(ABC) Trong mp ( A B C ) , gpi H = M N n B C , c6: ^ ' ^He(SAH)n(ABC) (2) |HeBC,BCe(BCD) T u (1) va (2) (SAH) n (ABC) = A H . ^^W.(BCD)n(MNO) (l) mnl^A.^^a Vaco 0 € ( B C D ) n ( M N O ) (2) ^ Gpi l = M N r ^ A H ( v i I V I N , A H c ( S A H ) ) . T u (1) va (2) suy ra ( B C D ) n ( M N O ) = H O . ,j *T b) Tim giao diem cua D C , BD voi (OMN). • —
leMN Co: I = MNn(ABC) HelJ •.'0.::/ .{\\ le AH,AHc(ABC) Co: • H = IJo(ABC) H€MN,MNCZ(ABC) Tim giao tuyen cua (MNP) va (ABC). b. Tim giao tuyen ciia mp(IJK) va mp(ABC). : , Taco: P e ( M N P ) n ( A B C ) (3) Ke(lJK)n(ABC) HK = (lJK)n(ABCJ Va I e (MNP) n (ABC) (4) H€(lJK)n(ABC) Tu (3) va (4) =^ (MNP) n (ABC) = P I . V:.//- GQi D = H K n B C ( H K , B C c ( A B C ) ) ; b) . Tim giao diem ciia AB vai (MNP). E = HK n A C ( H K , A C c ( A B C ) ) . GQi K = A B n P l ( v i A B , P I c ( A B C ) ) . Tim giao tuyen ciia mp(IJK) va mp(SBC). KeAB Co J € (IJK) n (SBC) (1) Co K = ABn(MNP). K€PI,PIc(MNP) ^ JDeHK,HKc(lJK) c) . Tim giao diem ciia NP vai (SAB). ^ |DeBC,BCc(SBC) G
P H l / O N G PHAP CHLfNG MINH BA D l / O N G T H A N G D O N G QUY: Cho hinh chop S.ABCD c6 day ABCD la hinh binh hanh tarn O. Goi M , N , I Ta tim giao diem cua hai duong thang trong ba duong thang da cho, roi Ian lugt nam tren ba canh AD, CD, SO. Tim thiet di§n ciia hinh chop voi mat chung minh giao diem do nam tren duong thang thii ba. Cu the nhu sau: phang (MNI). Cach chung m i n h 3 duang t h i n g a, b, c dong quy tai mot diem, f LOI GIAI Chon mot mat p h i n g (P) chua duong thang (a)va ( b ) . G o i I = ( a ) n ( b ) Trong (ABCD), goi: ] = BD n M N , K = MNnAB, H = MNnBC ' Tim mot mat p h i n g (Q) chua duong thang (a), tim mot mat phang (R) Trong (SBD), goi Q = IJ n SB chua duong t h i n g (b),saocho (c) = ( Q ) n ( R ) : Trong (SAB), goi K = KQ n SA Vay: 3 duong t h i n g (a),(b),(c) dong quy tai diem I . Trong (SBC), goi P = Q H n SC Vay : thiet dien la ngu giac MNPQR (a),(b)c:mp(P) (a)n(b) = I m p ( P ) n m p ( Q ) = (a) ^ ( a ) n ( b ) n ( c ) = I mp(P)nmp(R)-(b) mp(Q)nmp(R)=:(c) Cho hinh chiSp S.ABCi"). Goi M, N , I ' Ian liiot la tnmg diem lay tren AB Cho t u dien S.ABC. Tren SA, SB, va SC Ian luot lay cac diem D, E, F sao cho A D V a SC . DE cit AB tai I , EF cit BC tai j , FD cit CA tai K. Chiing minh ba diem I , J, K Tim thiet dien ciia hinh chop voi mat phang (MNP) thang hang^ LOI GIAI LOT^GIAI Trong (ABCD), goi: I = A B n D E ( AB,DE c ( S A B ) ) E = M N n DC, F = MN BC Co: AB,ABe(ABC) Trong (SCD), goi Q = EI' n SD l6DE,DEc(DEF) Trong (SBC), goi R = FP,",SB Ie(ABC)n(DEF) (l) Vay : thiet diC-n la ngu giac MNPQR K-ACnDF (AC,DFC(SAC)) Co Ke A C , A C C ( A B C ) D A N G 4: Chung minh ba diem t h i n g hang, ba duang t h i n g dong qui, KeDF,DFc:(DEF) chung minh mpt diem thupc mgt duang t h i n g co d i n h ^ K6(ABC)n(DEF) (2) Phuang phap: * ^' Muon cliiing minh ba diem A, B, C thang J = BCnEF(BC,EFc(SBC)) hang, ta chirng minh ba diem do Ian krot Co ]€BC,BCc(ABC) =^Je(ABC)n(DEF) (3) thupc hai mat phang phan biet (a) va (p), JeEF,EFc(DEF) thi suy ra ba diem A , B, C nam tren giao T u (1) (2) va (3) suy ra 3 diem I , J, K t h i n g hang. tuye'n ciia (a) va ((i), nen chung thang hang. 29
Cho t i i di#n ABCD c6 G la trpng tam tam giac BCD. Gpi M , N, P km iugt b) Xac djnh giao diem J = M N o (SBD) la trung diem ciia AB, BC, CD. • Chpn mp phu (SMC) chua M N . a) Tim giao tuyen ciia (AND) va (ABP). nvtib-.i- Tim giao tuyen ciia (SMC ) va (SBD): b) Goi I = AG n MP, ] = CM n A N . Chung minh D, I , J t h i n g hang. Trong (ABCD) goi K = MC n BD. . ^ , / , LOIGIAI Hai mat phiing (SMC) va (SBD) c6 hai a) . Tim giao tuyen cua (AND) va ( A B P ) . diem chung la S va K. Vay: (SMC) n (SBD) = SK A£(ABP)O(ADN)(I) • Trong (SMC), goi J = M N n SK . Taco G = B P n D N , c 6 : Vay J = M N n ( SBD) . ^ l;:J*ai/ JGeBP,BPc(ABP) c) Chung minh I , J , B t h i n g hang g I [GeDN,DNc(ADN) Ta CO : B la diem chung cua (ANB) va ( SBD) ; =>Ge(ABP)n(ADN)(2) l€SO,SOc(SBD) B V ,=5l6(SBD)n(ABN) T u (1) va (2) A G = ( A B P ) n ( A D N ) le AN,ANc(ABN) ^ ' ^ .^^ ' ^ b) . Chung minh D, I , J thang hang j6SE,SEc(SBD) Je(SBD)n(ABN) (3) I = AGoMP, AGc(ADG), MPC(DMN) ' JeMN,MNc(ABN)' =^l€(ADG)n(DMN) (3) T u (]) (2) (3) suy ra ba diem B , I , J thang hang. i ^ J = C M n A N , A N c ( A D G ) , CM c ( D M N ) => J e ( A D G ) n ( D M N ) (4) Cho tu giac ABCD va S g (ABCD). Gpi I, J la hai diem tren A D va SB, A D De(ADG)n(DMN) (s) rtMii^^ cat BC tai O va OJ c i t SC tai M . a) Tim giao diem K = IJ n (SAC) Tu (3), (4), (5) SLiy ra ba diem D, I J thuoc giao tuyen cua hai mat phing if') b.) Xac djnh giao diem L = DJ n (SAC) (ADG) va (DMN). c) Chung minh A , K , L , M thang hang Ket luan vay ba diem D , I , J thang hang. ' LOI GIAI Cho hinh binh hanh ABCD. S la diem khong thuQc (ABCD), M va N Ian a) Tim giao diem K = IJ n (SAC) lugt la trung diem cua doan AB va SC. • Chon mp phu (SIB) chua IJ a) Xac djnh giao diem I = A N n (SBD) . v • Tim giao tuyen cua (SIB ) va (SAC) b) Xac djnh giao diem J = M N n (SBD) it] Se(SBl)n(SAC) (l) c) Chung minh I, ], B thcing hang. Trong (ABCD) gpi E = AC n B I , c6: LOI GIAl EeACc(SAC) a) Xac dinh giao diem I = A N n (SBD ) . '' Ee(SAC)n(SBl) • Chon mp phu (SAC) chua A N . ' , y\ E € BI c (SBI) Tim giao tuyen cua (SAC ) va (SBD): T u (1) \'a (2) (SBI) n (SAC) = SE . Ql^Z) Trong mat phang (ABCD) gpi O la giao diem ciia AC va BD. Hai mat p h i n g Trong (SIB) gpi K = IJ n SE . (SAC) va (SBD) c6 hai diem chung la S va O. -AA } 'o iXtpii \ KelJ Vay: ( SAC) n (SBD) = SO i Ta CO • K = IJn(SAC) K€SE,SEc(SAC) • Trong (SAC) goi I = A N n SO. Vay: I = A N n ( SBD) '[ 31
b) X a c d i n h giao d i e m L = DJ o (SAC) • C h p n m p p h u (SBD) chua DJ . T i m giao tuye'n ciia (SBD ) va (SAC) leBN Ta c6: • • I-BNo(SAC) S e (SBD) n ( S A C ) (3) ) fiv(' Jl/rl) l€SO,SOc(SAC) V T r o n g ( A B C D ) goi F = A C n BD . • " b) T i m giao d i e m J = M N r\ SAC) Vay F la d i e m c h u n g t h u hai cua hai mat phMng (4) • ' • C h p n m p p h u ( S M D ) chua M N . ' T i r ( 3 ) v a ( 4 ) : (SBD) n ( SAC) = SF Tim giao tuye'n cua ( S M D ) va (SAC) T r o n g (SBD) goi L = DJ n SF . Trong (ABCD), goi K = AC n DM. - • ' ' ' ' • ' / H a i m a t p h 3 n g (SAC) va (SMD) c6 Ta CO : hai d i e m c h u n g la S va K . Lcsr,snc(SAC)=''-:°"^J = M N n ( S A C ) JK€lj,IJc„(AJO) JeSK,SKc(SAC) :>KG(SAC)n(AJO) V," KGSE,SEC(SAC) c) C h i i n g m i n h C, I , J thang hang f-l LeDJc(A]0) Theo each t i m d i e m 6 n h u n g cau tren ta c6 ba d i e m C, I , J la d i e m c h u n g ciia Co LG(SAC)n(AJO) (5) hai mat p h a n g ( B C N ) va (SAC) => Ba d i e m C, I , J c i i n g thupc giao tuyen ciia LeSFc(SAC) hai m a t phcing (BCN) va (SAC). Ket luan C, I , J thiing hang. M6jOc(AJO) Cho h i n h chop S.ABCD. Goi M , N , P Ian l u o t la t r u n g d i e m ciia SA, SB, Co ^MG(SAC)n(AJO) (6) M G SC c ( S A C ) SC. Goi !-: = A B n C D , K = A D n B C . a) T i m giao tuye'n cua ( S A C ) n ( S B D ) , ( M N P ) n ( S B D ) . T u ( 3 ) , ( 4 ) , ( 5 ) , ( 6 ) siiy ra bcVn diem A, K, L, M c i m g thuoc giao tuye'n ciia hai mat p h a n g (SAC) va (AJO). b) T i m giao d i e m Q cua d u o n g thang SD v o i m a t phang ( M N P ) . Vay: A, K, M thang hang. c) G o i H = N M n P Q . C h u n g m i n h 3 diem S, H , E t h ^ n g hang. • - ^ Cho tiV giac A B C D va S i (ABCD). Go! M , N la hai d i e m tren BC va SD. d) C h u n g m i n h 3 d u o n g thang SK, Q M , N P d o n g q u i . a) T i m giao d i e m I = B N n ( SAC) LOI GIAI b) T i m giao d i e m J = M N n ( SAC) rix:) /t j / • •» a). T i m giao tuye'n cua ( S A C ) n ( S B D ) . c) C h i r n g m i n h C , I , J thang hang / o T r o n g m p ( A B C D ) gc?i O = A C n B D , c6: OG AC,ACe(SAC) LOI GIAI OGBD,BDC(SDB) a) T i m giao d i e m I = B N n ( S A C ) ,„ K ; ; . • •; i •I =>OG(SAC)n(SBD) (l) • C h o n m p p h u (SBD) chua B N • T i m giao tuyen cua (SBD ) va (SAC) SG(SAC)n(SBD) (2) T r o n g ( A B C D ) goi O = A C n BD. T i l ( 1 ) va (2) ( S A C ) n ( S B D ) = S O . / H a i m a t phang (SAC) va (SBD) c6 hai d i e m c h u n g la S va O. V ^ y giao tuye'n T i m giao t u y e n ciia ( M N P ) n ( S B D ) cua c h i i n g la SO. U= , t,l • T r o n g (SBD), gpi I = B N n SO. T r o n g m p ( S A C ) goi F = M P n S O , co: , i riiffrt
rFeMP,MPc:(MNP) , , , Trong ASDFCO I M la d u a n g ••"->•' ' A ' j f b -jifi it I t / , ^=>Fe(MNP)n(SBD) (3) t r u n g b i n h c u a tarn giac. [FeSO,SOc(SBD) ^ / V ; V; /, ^ => S I = I = F A . N€(MNP) , , , ^ , , i . twvn,pf„;T Tu d o s u y ra F D la d u o n g Co: NeSB,SBc(SBD) ^ ' ^ ' ' t r u n g b i n h cua t a m giac A I K . => D t r u n g d i e m ciia A K . Tu(3)va(4) (MNP)n(SBD) = NF. ^" A ) ;:5noTT KD b) T i m giao d i e m Q cua d u o n g thSng SD v o l m $ t phSng ( M N P ) . - 5 ' ' ' '^'''^ Ket l u ^ n KA 2 G
b) M va I la hai diem chung cua hai mat phSng (MPQ) va (ACD). A e (SAC) n ( A M N ) (l) Vay giao tuye'n cua (ACD) va (MPQ) la duong t h i n g M I . EeSO,SO