Tài liệu Tuyển Tập 500 Bài Toán Hình Không Gian Chọn Lọc (NXB Đại Học Quốc Gia) - Nguyễn Đức Đông

516.23076 BAN GIAO VIEN NANG T527T N J G U Y E N KHIEU TRUCiNG THI eC/C e O N G ( C h u bien) PHAN LOAI VA PHl/OfNG PHAP GIAITHEO CHUYEN DE • BOI Dl/dNG HQC SINH GIOI • CHUAN B! THI TU TAI, DAI HOC VA CAO BOG Ha NOI DANG NHA XUAT BAN OAI HOC QUOC GIA HA NOI BAN GIAO V I E N NANG K H I E U TRl/CfNG THI NGUYEN DLfC D 6 N G {Chu hien) TUYEN TAP 500 BAITOAN • HDIH imm GIAN C H O N LOG • • • PHAN LOAI VA PHUdNG PHAP G I A I THEO 2 3 CHUYEN • B o i difdng hoc s i n h g i o i • C h u a n b i t h i T i i t a i , D a i hoc v a Cao d a n g (Tdi ban idn thvt ba, c6 svCa chUa bo sung) THir ViEN TiiVH BiKH liik^m NHA XUAT BAN DAI HOC QUOC GIA H A NOI NHA XUAT BAN DAI HOC QUOC GIA HA NQI 16 Hang Chuoi - Hai Ba Trcfng - Ha Npi Dien thoai: Bien tap - Che ban: (04) 39714896 Hanln chinli: (04) 39714899; Tong Bien tap: (04) 39715011 • Fax: (04) 39714899 * Chiu Gidm Bien Saa trdch ** nhiem xuat ban: doc - Tong bien tap: T S . P H A M T H I T R A M tap: THUY bdi: THAI Che ban: Trinh HOA VAN N h a sach H O N G A N bay bia: THAI V A N SACH LIEN K E T TUYEN TAP 500 BAI TOAN HJNH KHONG GIAN CHON LOG Ma so: 1L - 195OH2014 In 1.000 cuon, kho 17 x 24cm tai Cong ti Co phan V3n hoa VSn Lang - TP. Ho Chi IVlinh. So xuat ban: 664 - 2014/CXB/01-127/OHQGHN ngay 10/03/2014. Quyet dinh xuat ban so: 198LK - TN/QO - NXBOHQGHN ngay 15/04/2014. in xong va nop IIAJ chieu quy il nSm 2014. LCilNOIDAU Chung t o i x i n g i d i t h i $ u den doc gia bp sdch: Tuyen t a p cdc b ^ i toan d k n h cho hoc sinh Idp 12, chuan b i t h i vao cac trucrng D a i hoc & Cao d i n g . Bo sach gom 7 quyen : . T U Y E N T A P 546 B A I T O A N T I C H P H A N . T U Y E N T A P 540 B A I T O A N K H A O S A T H A M SO . T U Y E N T A P 500 B A I T O A N H I N H G I A I T I C H . T U Y E N T A P 500 B A I T O A N H I N H K H O N G G I A N . T U Y E N T A P 696 B A I T O A N D A I SO • T U Y E N T A P 599 B A I T O A N L U O N G G I A C . T U Y E N T A P 6 7 0 B A I T O A N RCJI R A C V A C l / C TRI NhSm phuc vu cho viec r e n luyen va on t h i vao D a i hoc b k n g phucrng phdp t i m hieu cac de t h i dai hoc da ra, de tiT n a n g cao va chuan b i k i e n thiJc can t h i e t . De phuc vu cho cac do'i tUcfng t\i hoc : Cac bai g i a i luon chi t i e t va ddy d u , p h a n nho tCrng loai toan va dua vao do cac phucfng phap hop l i . Mac du chiing t o i da co g^ng het siic t r o n g qud t r i n h bien soan, song vSn k h o n g t r a n h k h o i nhiJng t h i e u sot. Chiing t o i m o n g don n h a n m o i gop y, phe b i n h tii quy dong nghiep ciing doc gia de Ian xuat ban sau sach ducfc hoan t h i e n hcfn. Cuoi Cling, chiing toi x i n cam cm N I l A X U A T B A N D A I H O C Q U O C G I A H A N O I da giiip da chiing t o i m o i m a t d l bo sach dUdc r a dcfi. NGUYEN DtfC DONG 3 • (i) B A N G K E CAC K I H I E U V A CHLf V I E T T A T T R O N G • [ ( A B C ) ; ( E F G ) ] : goc tao bori 2 mp ( A B C ) va ( E F G ) -> • C > : Phep t i n h tien vectcf v V • D A : Phep doi xOmg true A • Do : Phep doi xiiTng true 0 • Q(0; cp) : Phep quay t a m O, goc quay (p. • V T ( 0 ; k ) : Phep v i t u t a m 0, t i so k. • D N : dinh nghla • D L : dinh ly • Stp : D i e n t i c h t o a n p h a n : The t i c h • C M R : chiJng m i n h r i n g A : goc • B i : budc i • T H i : t r u d n g hop i • V T : ve t r a i xuong dtfcfng thftng (d) (3r3^ SACH CAC K I H I E U T O A N HOC v A CAC T l / V I E T T A T <=> : (i) tUcfng dUcfng (il • => : (i) keo theo • : k h o n g tUdng dilcfng • d> : k h o n g keo theo • = : dong n h a t : k h o n g dong n h a t • i • Sv\nc = S ( A B C ) = d t ( A B C ) : d i e n . t i c h AABC • V s A H c = V ( S . A B C ) : the t i c h h i n h chop S.ABC • H Q : he qua • Sxq : D i e n t i c h xung quanh • V • A ' = ''7(ai A : A ' la h i n h chieu ciia A xuong m a t p h i n g (a) • A ' = ''Vfd) A : A ' l a h i n h chieu cua • d [ M ; (D)l : k h o a n g each tiT d i e m M d e n ducfng t h i n g (D) • d [ M ; ( A B C ) I : k h o a n g each tii diem M den mat phang ( A B C ) • (a; P ) : goc n h i d i e n tao bcfi 2 mfa m a t phang (a) va ( P ) • ( S ; A B ; D) = ( A B ) : n h i dien c a n h A B • tao bdi h a i dUomg t h i n g d • V P : ve p h a i • B D T : bat d i n g thijfc • y c b t : yeu cau b a i toan • d p c m : dieu p h a i chuCng m i n h • gt : gia thiet • K L : ket luan • D K : dieu k i e n • P B : phan ban va d' • [ H T C A B C T I : goc tao bdi du&ng t h i n g d va • C P B : chiTa p h a n ban mp(ABC) 4 Chuyen de 1 : TONG QUAN V E C A C KHAI NIEM T R O N G HINH H O C K H O N G G I A N • H i n h hoc k h o n g gian la m o t mon hoc ve cac v $ t t h e t r o n g k h o n g g i a n ( h i n h h i n h hoc t r o n g k h o n g gian) ma cac d i e m h i n h t h a n h nen v a t the do t h u d n g thiTcrng k h o n g ciing n f t m t r o n g mot m a t phang. • N h i f vay ngoai d i e m v a d i i d n g t h d n g k h o n g drfoTc d i n h n g h i a nhiT t r o n g h i n h hoc phAng; mon h i n h hoc k h o n g g i a n con xay di/ng t h e m mot doi tuong can n g h i e n ciifu nCfa la k h a i n i # m m g t p h a n g c u n g k h o n g difoTc d i n h n g h i a . K h i noi tori k h a i n i e m nay t a lien tuang den m o t m a t ban b a n g phang, m o t m a t ho nildc yen l a n g , m o t tb giay dat d i n h sat t r e n mot m a t da di/gc l a m phang.... No duoc k y hieu b d i cac chCf i n L a T i n h n h a : (P), (Q), (R), ... hoac cac chCf t h u d n g H y L a p nhU (a), ((5), (y), .... • M a t phang k h o n g ducfc d i n h n g h i a qua mot k h a i n i e m k h a c ; n h i f n g thifc te cho thfi'y mSt ph&ng CO nhutng t i n h chat cu t h e sau, goi la cac t i e n de : O T I E N D E 1: C o i t n h a t b o n d i e m t r o n g k h o n g g i a n k h o n g t h ^ n g h a n g (nghia la luon luon c6 i t n h a t 1 d i e m d ngoai m o t m a t p h ^ n g tiiy y). O T I E N D E 2: N e u m p t dtfdng th&ng v a m p t m a t p h ^ n g c 6 h a i d i e m c h u n g t h i dUcTng th&ng a y se n S m t r p n v ^ n t r o n g m a t p h a n g n e u t r e n . O T I E N D E 3: N e u h a i m a t p h & n g c 6 d i e m c h u n g t h i c h t i n g c 6 v 6 so' d i e m c h u n g : n e n h a i m a t p h S n g do c S t n h a u t h e o m p t d U d n g t h ^ n g d i q u a v 6 so' d i e m c h u n g a y . Di/cfng t h a n g ay goi la giao tuyen cua h a i m a t ph^ng. O T I E N D E 4: C o m p t v a c h i m p t m $ t p h a n g d u y n h a t d i q u a b a d i e m p h a n b i # t khong th^ng hang. O T I E N D E 5: T r e n m p t m § t p h a n g t u y y t r o n g k h o n g g i a n c a c d i n h l y h i n h h o c ph&ng scf c a p (da hoc tCr Idp 6 den Idp 10 va cac d i n h l y n a n g cao) d e u d i i n g . O T I E N D E 6: M o i d o a n th&ng t r o n g k h o n g g i a n d e u c 6 dp d a i x a c d i n h : t i e n de neu len sU bao toan ve dp dai, goc va cac t i n h chat lien thuoc da biet t r o n g h i n h hoc p h i n g . • TiT do chung t a c6 m o t so each xac d i n h m a t p h 4 n g n h i / sau : O H E Q U A 1: C o m p t v a c h i m p t mfit p h S n g d u y n h a t d i q u a m p t d U d n g t h S n g v a m p t d i e m n S m n g o a i dt^dng t h a n g do. O O H E Q U A 2: C o mpt v a c h i mpt m^t p h d n g duy n h a t d i q u a h a i di^cAig t h ^ n g cSt n h a u . H E Q U A 3: C o m p t v a c h i m p t m ^ t p h a n g d u y n h a t d i q u a h a i di^c/ng t h d n g song song. • Dong t h d i t a phai hieu t h e m r k n g mot m a t phang se r o n g k h o n g bien gidi va dUcmg t h ^ n g c6 do dai v6 t a n mac du t a se bieu dien no mpt each h i n h thiifc hflu h a n va k h i e m t o n nhU sau: • De thuc h i e n dirge phep ve c h i n h xdc m 6 t h i n h h i n h hoc t r o n g k h o n g g i a n ngoai cac dudng t h a y ve l i e n n e t , t a can p h a i n a m chac di/pc k h a i n i e m di/dng k h u a t ve b k n g net dijft doan: Mpt dtfdng b i k h u a t t o a n bp h a y c h i k h u a t m p t d o a n c u e bp n a o do k h i v a c h i k h i t o n t a i i t n h a t m p t m a t p h S n g du'ng p h i a trvC6c h o ^ c p h i a t r e n c h e n o m p t e a c h t o a n bp h o a c c u e bp ti^cAig uTng. 5 • Muon xac d i n h n h ^ n h m o t m a t p h ^ n g t r o n g k h o n g gian t a con chon t h u thuat thUc h a n h : M p t h i n h t a m g i a c , tii" g i a c h o a c d a g i a c ph&ng ( k h o n g g e n h ) , dUcfng i r o n , l u d n x a c d i n h m p t m ^ t p h S n g t r o n g k h o n g g i a n . T a gpi c a c m&t p h ^ n g do l a m^it p h S n g h i n h thvCc v d i c a c k y h i p u ( A B C ) , ( A B C D ) , ( C ) , ... txictng vtng. M p t dvictng t h d n g n ^ m t r o n g m ^ t p h & n g h i n h thd'c m a m a t do h i k h u a t c u e bp • M a t p h d n g h i n h thu^c h i k h u a t n e u c 6 m p t h a y n h i e u m ^ t ph&ng n a o do c h e n o . • h a y t o a n bp v a k h i dUcTng t h ^ n g do k h o n g l a b i e n c u a m a t p h d n g b i k h u a t do, t h i di^dng th&ng do c u n g tii'oTng vlng k h u a t c u e bp h a y t o a n bp. Noi h a i d i e m m a it n h a t c 6 mpt d i e m k h u a t t h i dUpc mpt dUcfng k h u a t cue bp h a y • Mpt d i e m nhm t r o n g m p t m $ t ph&ng h i n h thuTc b i k h u a t t h i goi l a d i e m k h u a t . • t o a n bp : n e u h a i diictag do k h o n g l a b i e n c u a c a c m^t phAng h i n h thufc c h e no. • C A C H I N H A N H M I N H HQA \(d) • (d) b i (a) che k h u a t cue bo, do (d) c6 1 doan ve net dijft doan n k m dudi (a). S • (d) b i m a t p h ^ n g (SAC) che k h u a t cue bo, do (d) CO m p t doan ve duft doan n k m sau (SAC) (hien n h i e n (d) cung d sau cac m a t (SAB), (SBC)). • C a n h AC b i h a i m a t p h a n g (SBC) v£l (SBC) che k h u a t toan bo, do ca doan AC x e m n h u hoan t o a n d sau dong t h d i h a i m a t p h ^ n g (SAB), (SBC). -AA. c./—1—^VFJL^ • • A ] H b i che t o a n bo do ca doan A ] H n k m sau m a t p h i n g ( A i A D D i ) , mSc dij no d trU H a i m a t p h l n g (a), (P) thuf tif chiJa h a i difdng t h i n g ( d i ) , (da) ma (dj) n (da) = I => S I la giao tuyen can t i m . > H a i m a t p h l n g (a), (P) thuf t i f chtifa h a i difdng t h i n g ( d i ) , (da) ma ( d i ) // (da). S_ D i f n g xSy song song v d i (dj) h a y (da) => xSy la giao t u y e n can t i m . 7 m. C A C B A I T O A N C O B A M Bai 1 Cho tiif giac l o i A B C D c6 cac canh doi k h o n g song song va d i e m S d ngoai (ABCD). T i m giac tuyen ciia : a/ (SAC) va (SBD). hi (SAB) va (SDC); (SAD) va (SBC). Giai a/ Xet h a i m a t p h a n g (SAC) va (SBD), t a c6 : T r o n g tuT giac l o i A B C D , h a i ducmg cheo A C • S la d i e m c h u n g thuf n h a t . • (1) n B D = O : d i e m c h u n g thijf n h i (2). ^ Ti/(1) va (2) suy r a : (SAC) o (SBD) = SO (ycbt) hi Xet hai m a t p h a n g (SAB) va (SDC) cung c6 : H a i canh ben A B va C D cua t i l giac A B C D • S la m o t d i e m chung. • theo gia t h i e t k h o n g song song. ^ A B ^ C D = E : la d i e m c h u n g thut h a i . Do do : (SAB) n (SDC) = SE (ycbt) Tucfng t i f : (SAD) n (SBC) = SF (ycbt); v d i F = A D ^ BC; do A D / / BC. Bai 2 Cho t i l d i e n A B C D . Goi G j , Ga la t r p n g t a r n h a i t a m giac B C D va A C D . L a y theo thuT t i i I , J , K la t r u n g d i e m ciia B D , A D , C D . T i m cac giac tuyen : aJ (G1G2C) o ( A D B ) hi (G1G2B) n ( A C D ) c/ ( A B K ) o (CIJ>. a/ (G1G2C) n ( A B D ) = I J (ABK) ^ (CIJ) = d (GiGaB) n ( A C D ) = GgK hoSc A K hi G,G2 Bai 3 Cho h i n h chop S . A B C D c6 day A B C D la h i n h b i n h h a n h t a m O. T i m giao t u y e n cua h a i mSt p h i n g (SAB) va (SCD). hi T i m giao t u y e n cua h a i m a t phSng (SAD) va (SBC). aJ c/ T i m giao t u y e n ciia h a i m a t p h ^ n g (SAC) va (SBD). Giai aJ Xet h a i m a t phSng (SAD) va (SBC), t a c6 : De y A D c ( S A D ) ; BC c (SBC) m a A D // BC. • S la d i e m c h u n g thur n h a t . • Ta d u n g xSy // A D hoac BC. [(SAD) = (xSy; AD) ^ |(SBC) = (xSy; BC) =^ (SAD) n (SBC) = xSy (ycbt). hi Tifang t i r , difng uSv // A B hoftc C D 8 => (SAB) r^ (SCD) = uSv (ycbt) c/ Goi O = A C n B D , tiTcrng t a b a i 1 => (SAC) n (SBD) = SO (ycbt). Bai 4 Cho h i n h chop S . A B C D c6 day la h i n h t h a n g A B C D v d i A B l a day Idtn. Gpi M la m o t d i e m bat ky t r e n SD va E F l a difang t r u n g b i n h cua h i n h t h a n g . a/ T i m giao t u y e n ciia h a i mSt p h i n g (SAB) va (SCD). b/ T i m giao t u y e n cua h a i m a t phSng (SAD) va (SBC), c/ T i m giao t u y e n cua h a i mSt p h a n g ( M E F ) va ( M A B ) . Doc gia t u g i a i tUcfng t u n h u cac b a i t r e n . Bai 5 Cho h i n h chop S . A B C D c6 A B C D l a h i n h b i n h h a n h . Goi G,, G2 l a t r o n g t a m cac t a m giac SAD; SBC. T i m giao t u y e n cua cac cSp mSt p h a n g : a/ (SGiG^) va ( A B C D ) b/ (CDGiGz) va (SAB) UvCdng 0/ (ADG2) va (SBC). d§Ln Goi I , J , E, F thur t a Ik t r u n g d i e m cac doan t h i n g A D , BC, SA, SB theo thur tvt d6. Thifc h i e n cac l a p l u a n nhtf cac bai toan t r e n ; a/ (SG1G2) n ( A B C D ) = I J (ycbt) b/ (CDGiGa) n (SAB) = E F (ycbt) c/ (ADG2) ^ (SBC) = xG2y (ycbt) T r o n g do xGay // A D hoSc BC. L o a i 2 : T l M G I A O D I £ M C U A D U d N G T H A N G 1fA M A T L PHirONG PHANG PHAP Ca sd cua phaang phap t i m giao d i e m O cua dudng t h a n g (a) va m a t phSng (a) l a xet 2 h a i k h a nSng xay r a : n T r i r d n g hop (a) chiJa dudng t h S n g (b) va (b) l a i c&t diicrng t h d n g (a) t a i O. T i m O = (a) n (b) => O la d i e m can t i m . n Trtfdng hap (a) k h o n g chiifa dUcmg t h i n g nao cat (a). T i m ( P ) ^ ( a ) v a ( a ) n ( P ) = (d) > T i m O = (a) o (d) => O la d i e m can t i m . n. CAC BAI TOAM G O B A N Bai 6 Cho tuf d i e n A B C D . Goi M , N I a n lugt la t r u n g d i e m cua A C va BC. L a y d i e m K e B D sao cho K B > K D . T i m giao d i e m ciia h a i dudng t h i n g CD va A D v d i ( M N K ) . 9 • De y den K B > K D Do do t r o n g (BCD) Ma K N c ( M N K ) • Giai => K N k h o n g song song C D K N o CD = I . CD ( M N K ) = I (ycbt) Taong t a xet I M c ( M N K ) , t r o n g ( A D C ) Ta CO : AD n IM = E => A D n ( M N K ) = E (ycbt) Bai 7 Cho tiJ dien A B C D . L a y d i e m M t r e n A C va h a i d i e m N va K thuf tiT nSm t r o n g cac t a m giac B C D va A C D . D u n g giao d i e m cua CD va A D \di ( M N K ) . HtfdTng d i n Doc gia t u g i a i , x e m h i n h ben. a/ CD ( M N K ) = P (ycbt) b/ A D n ( M N K ) = Q (ycbt) Bai 8 Cho h i n h chop tuf giac S.ABCD. L a y t r e n SA, SB va BC ba d i e m M , N , P theo t h i i t\i sao cho M P k h o n g t h e c&t A B hay C D . T i m giao d i e m cua SC va A C v d i ( M N P ) . Giai ThUdng t h u d n g do ycbt t i m giao d i e m NP o SC = K ma NP c (MNP) Trong mp(SAC) ma M K c (MNP) SC n ( M N P ) = K (ycbt) M K o AC = H1 | => A C r> ( M N P ) = H (ycbt) Bai 9 Cho m o t t a m giac A B C va m o t d i e m S d ngoai m a t p h i n g chila t a m giac. T r e n SA va SB ta lay hai d i e m M , N v a t r o n g m a t p h i n g (ABC) ta lay mot d i e m O. D i n h ro giao diem cua ( M N O ) v d i cac dudng t h i n g A B , B C , A C va SC. Hi^dng d i n Tuang t u , doc gia t u g i a i (xem h i n h ben) A B n ( M N O ) = E (ycbt) BC o ( M N O ) = F (ycbt) A C n ( M N O ) = G (ycbt) SC n ( M N O ) = H (ycbt) 10 Loal 3 : Cfll/NG MWfl B A D I £ M T R O N G K H O N G G I A N T H A N G H A N G I. pmroNG P H A P Co so cua p h i i o n g phap can p h a i chufng m i n h ba d i e m trong yeu cau b ^ i t o a n l a d i e m chung cua 2 mSt phSng nao do (chfing b a n A, B, C nSm t r e n giao t u y e n (d) cua h a i m a t phSng do nen A, B, C t h a n g hang). O day k h o n g l o a i triJ k h a n&ng chiJng m i n h difoc difdng thang A B qua C => A, B, C t h i n g hang. n. C A C B A I T O A N C O BAM B a i 10 Xet ba d i e m A, B, C k h o n g thuoc m a t p h i n g (u). Goi D, E, F I a n l u o t l a giao d i e m ciia A B , EC, CA va (g). ChCifng m i n h D, E, F t h a n g hang. Giai De y t h a y D, E, F viTa a t r o n g m p ( A B C ) vifa d t r o n g mp(a). Do A, B, C g (a), nen (a) va (ABC) p h a n b i e t nhau. => ( a ) n (ABC) = A (A chuTa D , E, F) D, E, F t h i n g h a n g t r e n A (dpcm). B a i 11 H a i t a m giac A B C , A B C k h o n g dong p h i n g c6 A B n A B ' = I , A C n A C = J , BC n B C = K ChiJfng m i n h I , J , K t h i n g h a n g . Giai De y I , J , K I a n l u o t d t r e n h a i m a t p h i n g p h a n (P) ^ (ABC) va (Q) = ( A ' B ' C ) . N e n no 1^ diim c h u n g cua h a i m a t p h i n g do I , J , K e (A) = (ABC) n ( A ' B C ) => 1, J , K t h i n g h a n g (dpcm). B a i 12 Cho A, B l a h a i d i e m d h a i p h i a khac n h a u doi v d i m a t p h i n g a va A B c i t a t a i O. D i t o g hai dUdng t h i n g x'Ax, y ' B y song song n h a u theo thuf tiT c i t a t a i M va N . ChuTng m i n h M , N , O t h i n g hang. Kvldng TifOng ^ t\l: • fM, O, dSn NG(S) [8 = (Ax; By) n (a) => M , N , O t h i n g h a n g t r e n (8) 11 t o a l 4 : CmiUG MWfl M Q T DtfCiNG T H A N G T R O N G KHONG G I A N Q U A M O T D I £ M C O DINH I. PHirONG PHAP, Ca sd cua phucfng phap chuTng m i n h diXcrng t h i n g (d) qua m o t d i e m co d i n h : Ta can t i m t r e n (d) h a i d i e m tuy y A ; B va chuTng m i n h 2 d i e m do t h i n g h a n g v d i m o t d i e m I co d i n h c6 sSn t r o n g khong gian. => (d) qua I CO d i n h (dpcm). IL PHtfONG PHAP, Co sd cua phiTcfng phap can thuc h i e n ba bifdc ccf ban : n B i : T i m dUctng t h i n g a co d i n h d ngoai mSt p h 5 n g co d i n h (a) ma (a) chila d (liOi dong). • B2 : T i m giao d i e m I = a ^ d => I l a d i e m co d i n h ma d d i qua m. C A C B A I T O A N C O B A N Bai 13 Cho A , B l a h a i d i e m co d i n h t r o n g k h o n g g i a n d ve h a i p h i a khac n h a u cua m&t p h i n g co d i n h a. X e t d i e m M luu dong t r o n g k h o n g g i a n sao cho M A n a = I va M B n a = J . ChuTng m i n h difdng t h i n g I J luon d i qua m o t d i e m co d i n h . Giai Goi O = A B n (a) => O co d i n h ( v i A , B co d i n h vk a 2 p h i a cua (a)) T a CO : mp(P) = ( M A ; M B ) n (a) = I J De y t h a y : O e I J => O, I , J t h i n g hang. N g h i a l a dacfng t h i n g I J d i qua O co d i n h (dpcm) Bai 14 Cho h i n h t h a n g A B C D ( A B // C D va A B > CD). X e t d i e m S e ( A B C D ) va m a t p h i n g a luu dong quanh A C v d i a '-^ SB = M , a n SD = N . ChuTng m i n h difdng t h i n g M N luon luon d i qua mot d i e m co d i n h . De t h a y dxiac n g a y M N c (SBD) va AC c (SAC) va M N o A C = O t h i O e B D = (SBD) n (SAC) => M N qua O co d i n h (dpcm). 12 Bai 15 Cho h a i d i f d n g t h f t n g d o n g q u y O x , O y v a h a i d i e m A , B k h o h g n S m t r o n g m a t phing (xOy). M o t m a t p h a n g l i f u d o n g ( a ) q u a A B l u o n l u o n c a t O x , O y t a i M , N . C h i i f n g t o M N q u a mot d i e m co d i n h . Giai D e y t h a ' y k h i ( a ) q u a y q u a n h A B co d i n h n h t f n g vAn co : ( a ) n [ ( O x ; O y ) ^ (P)] = A ( q u a M , N ) 1 AB CO d i n h Nhung -j[P ABo(p) = I e A CO d i n h N g h i a la d u d n g t h a n g M N = A l o u dQng nhOng v a n qua I co d i n h . ( d p c m ) toal S : C H O N G MWfl B A O U d N G T H A « G T R O N G KHONG G I A N D O N G Q U Y L PHUONG PHAP, Co so cua p h i f a n g p h a p l a t a c a n c h i i f n g m i n h d U d n g thiif n h a t qua g i a o d i e m c i i a 2 d i f d n g c o n l a i b a n g 2 budrc co b a n : • • Bi : T i m (d,) o (d^) = O B2 : C h u f n g m i n h (d;j) q u a O . => ( d i ) , (d2), (d.i) d o n g q u y t a i O ( d p c m ) Q. P H U O N G PHAP, C o s d c u a p h a a n g p h a p l a t a c a n chijfng m i n h chung doi m o t cat n h a u v a d o i m o t d t r o n g 3 m a t p h a n g p h a n b i e t q u a 2 bifdc ca b a n : d], • B i : Xac d i n h < c: a; d j 0 da = I i id3 \ da, d;j cz P; da ^ d3 = I2 A dg, d , e Y; dg n d j = I3 di \ \ a, p, y p h a n bi§t • B2 : K e t l u a n ( d , ) ; (da); (d;,) d o n g q u y t a i 0 = I i = I2 = I3 m. c A c BAI TOAN C O BAM Bai 16 C h o t i l d i e n A B C D . G o i E , F , G l a b a d i e m t r e n b a c a n h A B , A C , B D sao c h o E F n B C = I , E G o A D = J ( v d i I ^ C wk J ^ B). C h i j f n g m i n h C D , I G v a J F d o n g q u y . Giai X e t b a difcfng t h A n g C D ; I G v a J F , t a tha'y : CD, I G e ( B D C ) va C D • IG, J F c (EFG) IG / 0 va I G n J F * 0 J F , C D e ( A C D ) va J F r> C D * 0 Va ba m a t p h a n g ( B C D ) , ( E F G ) , ( A C D ) l u o n p h a n b i e t ( v i I ?t c v a J ?t D ) => C D , I G , J F d o n g q u y t a i O ( d p c m ) . 13 O Cach khac D o c g i a churng m i n h r S n g J F q u a O = I G n C D => C D ; I G v a J F d o n g quy. B a i 17 C h o h a i t a m giac A B C , A B C sao cho A B c a t A ' B ' a E , A C cdt A C d F ; B C c a t B C d G . a/ Chufng m i n h b a d i e m E , F , G t h S n g h a n g . b/ C h i J n g m i n h difcfng t h a n g A A ' , B B ' , C C d o n g quy. Gidi a/ D e y t h a y E , F , G l a b a d i e m chung cua h a i m a t ph^ng p h a n biet (a) ^ ( A B C ) v a (P) = ( A B ' C ) . D o do : E , F , G e (A) = ( a ) n (P). V a y E , F , G th^ng h a n g (dpcm). b/ N h a n x e t n h u s a u : : AA', B B ' cr ( E A A ' ) ; A A ' o B B ' # 0 ^ B B ' , C C c ( G B B ' ) ; B B ' r^ C C * 0 Ice, ^ AA' c ( F C C ) ; C C n AA' # 0 A A ' , B B ' , C C d o n g quy t a i O (dpcm). Chuyen de 2 : QUAN HE SONG SONG t o a i 1: C H t J N G MWfl HAI DLfCJNG THANG SONG SONG I. PHirONG PHAP C o S0 c u a p h a o n g p h a p c a n t h i i c h i e n h a i hxidc CO b a n c h o d i n h n g h i a a // b j a , b c: (a) 'a^b = 0 • B i : K i e m t r a h a i difdng t h a n g a c u n g t r o n g m o t m a t p h a n g h a y hifeu n g a m r a n g h i e n n h i e n d i e u do x a y r a n e u c h u n g t r o n g 1 h i n h p h a n g n a o do. ( 1 ) • B 2 : D u n g d i n h ly T h a l e s , t a m giac dong dang, t i n h c h a t bac cau ( t i n h c h a t cung song s o n g \6i difdng thiJ b a ) l a h a i c a n h c u a h i n h t h a n g , h a y h a i c a n h doi c u a h i n h b i n h h a n h , ... de k h a n g d i n h h a i difcfng t h ^ n g do k h o n g c6 d i e m c h u n g . ( 2 ) T i f ( 1 ) v a ( 2 ) => ( y c b t ) n. C A C BAI TOAN CO BAN B a i 18 C h o h i n h c h o p S . A B C D c6 G j , G 2 , G3, G , I a n lucft l a t r o n g t a m c a c t a m g i a c S A B , S B C , S C D , S D A . C h u m g m i n h tiJf g i a c G i G a G g G , l a h i n h b i n h h a n h . 14 Giai SG, SE Theo t i n h chat t r o n g tarn, t a c6 : - i , SG3 t [ SH SG2 SF 2 SG4 2 SK 3 3 Dinh l y Thales va t i n h chat diTcfng t r u n g b i n h G,G2// = - E F ; E F 7 / = i A C ' ^ 3 2 • G1G2 // = G;jG4 G.G,, // = - H K ; HK// = - AC ^ ' 3 2 G1G2G3G4 l a h i n h b i n h h a n h (dpcm). B a i 19 Cho diem S d ngoai m a t phSng h i n h b i n h h a n h A B C D . X e t m S t p h d n g a qua A D c^t SB va SC Ian lucft d M va N . Chiirng m i n h A M N D l a h i n h t h a n g . Giai S D6 y thay h a i m S t phSng (a) v a (P) c6 2 d i e m M vfl N 1^ d i ^ m chung. => M N = (a) n (SBC) '(a) 3 AD ma^(SBC)3BC iAD//BC N va theo each d i m g M N // A D (hoftc BC) => A D N M l a h i n h t h a n g day lorn A D . (dpcm) B a i 20 Cho tuT dien A B C D . Goi M , N I a n li^gt l a t r u n g d i e m cua B C va B D . G g i P l a d i e m t u y y tren canh A B sao cho P ?t A v a P # B. X e t 1 = P D A N va J = PC o A M . ChiJng m i n h r S n g : I J // C D . Giai Xet h a i m a t p h a n g ( A M N ) v a (PCD) c6 h a i d i e m chung l a I va J . IJ = ( A M N ) r-. (PCD) 'CD c (PCD) N h i m g < MX CT (AMN) • va MN // CD ^ I J // M N hoac C D (dpcm). toai Z : CfltJfJG M W H DiidfiG T H A N G S O N G S O N G TfCl M A T F H A N G L PHtrOWG P H A P , Co so ciia phuong phap m o t l a sii dung d i n h l y phuong giao t u y e n song song. De chiing m i n h d // a t a can thUc h i e n h a i bade CO b a n chufng m i n h : • E l : Chufng m i n h d = y o p m a • B2 : K e t l u a n t i f t r e n d // a. d y r- a = a p n a = b. a//b 15 n . PHOONG PHAP^ Ca sd ciia phifcng phap la stf dung dieu k i e n can va du chijfng m i n h di/dng t h i n g (d) song song vcJi m a t p h a n g (a) b a n g h a i btfdrc : • B i : Quan sat va quan l y gia t h i e t t i m dudng t h i n g ou v i e t (A) cz (a) va chiJng m i n h (d) // (A). • B2 : K e t l u a n (d) // (a) theo dieu k i e n can va dii. m. cAc BAI T O A N C O BAM Bai21 T r o n g tuf dien A B C D , chufng m i n h rSng dean no'i h a i t r o n g t a m G i , G2 cua h a i A A B C va A A B D t h i song song v6[ ( A C D ) . Giai A Goi A i , A2 l a t r u n g d i e m BC va B D theo thut tiT do, t a c6 : AG2 3 AA, ' AAg 2 AG) Theo d i n h l y T h a l e s , t a c6 : ' 0 , 0 2 / / A , A2 B ' m a A,A2 //CD (tinh chat dUcrng trung binh) Theo t i n h bSc cau => G1G2 // CD c: (ACD) =j. G1G2 // (ACD) (dpcm) B a i 22 Cho h i n h chop S.ABCD day l a h i n h b i n h h a n h A B C D . G o i M , N l a t r u n g d i e m SA va SB. Chijfng m i n h : M N // (SCD) v a A B // ( M N C D ) . Giai Theo t i n h c h a t dudng t r u n g b i n h t r o n g t a m giac => M N // A B , ma A B // CD => M N // C D Theo dieu k i e n can va du O cz (SCD) => M N // (SCD) (ycbt). Cach khac De y M N = ( M N C D ) n (SAB) va t r o n g h a i m a t p h a n g do chiJa theo thijf tiT cac doan t h i n g C D // A B D => M N // (SCD) 3 CD (ycbt) M N // A B va C D TifOng tyl : A B // M N c ( C D M N ) => A B // ( C D M N ) (dpcm). B a i 23 Xet h a i h i n h b i n h h a n h A B C D va A B E F k h o n g dong p h l n g . Goi M , N l a h a i d i e m thoa AM - i AC va BN = - BF . Chufng m i n h r i n g M N // ( D E F ) . 3 3 Giai De y t h a y M , N l a t r o n g t a m cua b a i t a m giac A B D va A B E theo thijf t u do. Keo d a i t h i D M o E N = P : l a t r u n g d i e m A B . ^ PE PD PX PM 1 3 Theo d i n h l y T h a l e s ^ M N // E D c ( E F D C ) ^ ( D E F ) (dpcm) D 16 Bai 24 H i n h c h o p S . A B C D c6 d a y l a h i n h b i n h h a n h A B C D , t a r n O . G o i M , N I a n \\iqt l a t r u n g d i e m S A , S B v a x e t h e t h i J c v e c t o : 3 S I - 2 S M = 3 SJ - 2 S N = 0*. ChuTng m i n h r S n g : a/ I J / / ( S C D ) b/ S C / / ( M N O ) . Hvfdrng d i n a/ i I J // M N , M N // A B ; A B // C D M N // C D CD c (SCD) => I J / / ( C S D ) ( d p c m ) b/ AM AO AS AC S C // M O c ( O M N ) S C // ( O M N ) ( d p c m ) Bai 25 C h o A x , B y l a h a i nijfa d i T d n g t h S n g c h e o n h a u . T r e n A x l a y d i e m M , t r e n B y l a y d i e m N sao c h o A M = B N . C h i j f n g m i n h r S n g dUcfng t h i n g chufa d o a n M N l u o n l u o n s o n g s o n g w6i m a t p h a n g CO d i n h . Q u a A d u n g A x ' // B y ; q u a N d i f n g N N ' // B A ; v6i N ' e A x ' . L u c d o tii g i a c A N N B l a h i n h b i n h h a n h n e n : A N ' = B N => A M = A N ' De y A A M N ' c a n d A n e n t i a p h a n giac n g o a i A t cua STAJT se s o n g s o n g v6i M N ' v a t i a A t n a y co d i n h h a y A B v a A t x a c d i n h m a t p h S n g co d i n h ( P ) . Ta lMN'//At CO : < [ ( M N N ' ) // ( P ) N N ' // A B V a y : M N // ( P ) tiifc l a M N l u o n l u o n s o n g s o n g v6i m&t p h a n g co d i n h (dpcm). toal 3 : HAI M A T P H A N G S O N G S O N G Dang 1 : C H Q N G MINH HAI MAT P H A N G S O N G S O N G L PmrOHG PHAP Co sd cua phuong phap chiJng m i n h hai mat p h a n g fx v a P s o n g s o n g n h a u t a c a n thiTc h i e n h a i bUdc CO b a n t r o n g k h i siJf d u n g d i e u k i e n c a n v a d u nhu sau: • B i : Chufng m i n h " m a t p h a n g ( a ) c h i i a h a i dUcJng t h a n g a, b d o n g q u y thijf t i f s o n g song v d i h a i dUoing t h a n g a', b ' d o n g q u y t r o n g m a t p h a n g P". • B2 : K e t l u a n ( a ) // (P) t h e o d i e u k i e n c a n v a d u . THL; VJENTifJHglNHTHUAN 17 n . ckc BAITOAN C OBAM Bai 26 T r e n b a t i a c u n g c h i e u , s o n g s o n g v a Ichong d o n g p h ^ n g A x , B y , C z M n lifot l a y c a c d i e m A ' , B ' , C s a o c h o : A A ' = B B ' = C C c 6 do d a i k h a c k h o n g . ChOfng m i n h ( A B C ) // ( A B C ) . Giai AA' =3 BB' D e y : ( A B ' C ) // ( A B C ) ( d p c m ) Bai 27 C h o h i n h b i n h h a n h A B C D . Tir A v a C k e A x c a C y song song cung chieu v a khong n k m t r o n g m a t p h S n g ( A B C D ) . Chiifng m i n h ( B ; A x ) // ( D ; C y ) . Gi&i Tirang t u xet h a i m a t p h i n g ( B ; A x ) v a ( D ; C y ) , thuT t a chuTa c a c c a p d u d n g thing d o n g quy. fAB//CD IAx//Cy => ( B ; A x ) // ( D ; C y ) ( d p c m ) Bai 28 C h o h a i h i n h binh h ^ n h A B C D v a A B E F d trong h a i m a t ph^ng khac nhau. Chilng m i n h ( A D F ) // ( B C E ) . Giai H a i m a t p h l i n g ( A D F ) v a ( B C E ) thiif tiT chuTa c a c c a p dirdng t h d n g d o n g quy. iAF//BE AD//BC / A; ( A D F ) // ( B C E ) ( d p c m ) Dang 2 : CHUfNG MINH CAC Dl/dNG THANG D6NG PHANG LPBirONGPBAP Ccf s d c u a p h u a n g p h a p chiifng m i n h c a c d u d n g t h i n g d i , d2, dg... d o n g p h i n g l a c a n p h a i thiTc h i ^ n h a i bi/ d i , d2, d^j, ... dong p h i n g trong (a); (a) phai chufa cac giao diem cija d,, da, ds, .... n. C A C B A I T O A N C O B A N Bai 29 Cho tiJ dien ABCD c6 AB = AC = AD. Chufng minh rSng ba diTcfng phan giac ngo^i cdc goc SAC. CAI), I5AB cung nSm trong mot mat phlng. Giai Goi A t i , At2, Ata la ba diTdng phan giac ngoai ciia goc : fiAfc, CXt), I5A6 theo thuT t u do. Do cac tam gidc can tai dinh A nen cac phan giac ngoai song song vdi canh day, nen : At, / / B C c (BCD) A t a Z / C D e (BCD) ;At3//BDc(BCD) At,, At2, At3 // (BCD) => A t , , At2, Ata dong ph^ng (trong (P) // (BCD) \k (P) qua A) (dpcm). Bai 30 Cho hinh chop day la luc gidc deu. Chufng minh rang giao tuyen cua mat ben doi nhau thi dong phlng. Giai De y thay : (SAB) n (SED) = t, // AB, E D (SBC) o (SEE) = ta // BC, E E ^(SCD) n (SEA) = => // CD, FA t , , ta, tg//(ABCDEF) Vay t , , t2, tg dong ph^ng trong (a) // (ABCDEF) va (a) qua S. (dpcm) Bai 31 Tren bon tia phan biet Ax, By, Cz va Dt song song cung chieu, lay cac diem A', B', C , D' sao cho AA' = BB' = CC = DD'. Chutng minh r i n g A B , B'C, CD', D A ' , A C , B'D" cung song song vdi mat ph^ng ABCD. Htfdng d i n Doc gia t u giSi iMng t\l hai bai toan tren. 19