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Boi ditdiig IISG Ilinli hoc khdng ijian - Phan Buy CHl/t^NeJ Clij TNini lihni MTV DVVII Khnng Viet KHOITRONXOAY §1 i r i l W I C A U Cac b a i loan vc hinh cau hinli hoc Ichong gian ihutin thien vc c;ic tinh chat dinh tinh De giai du'dc cac bai toan phan d o i h o i hoc sinh phai nam vi^ng va su" dung thao cac kie'n thiirc cua hinh hoc khong gjan (dac biet la ciic kie'n thiJc ve quan he song song va quan he vuc)ng goc) ,ii - A T o m t a t ly t h u y e t V j t r i tU"dng d o i giffa hai mat cau - H i n h cau la tap hdp nhffng d i e m M khong gian ma khoang each ttr M Cho hai hinh cau S|(Oi; R|) va S:(02; R2) de'n mot d i e m co dinh O luon luon nho hcfn hay bang m o t dai R cho D a l d = | : la khoang each giffa hai lam cua hai hinh cau tru-dfc, ti?c la O M < R a " N e u d > R| + R: i h i hai hinh cau khong cat vii (If ngoai d > R| + R2 M a t cau la tap hdp nhiyng d i e m M each deu mot d i e m c6 dinh O m o t doan b Ne'u d = R| + R: thi hai hinh cau liep xiic ngoai v6i khong d o i bang R, tifc la O M = R • ,:, V i t r i tUcJng d o i gifra mat phring va mat c;1u X e t hinh cau S(0; R) tarn O, ban kinh R va mSt phang (P) Gia su" d la khoang each tif tam O de'n mat phang (P) a d = R| + R N e u d > R: M t i t phang (P) va mat ciiu S(0; R) khong cat c N e u |R| - R2I < d < R i + R thi hai hinh cau cat : • viU | R , - R i < d < R , + R2 d d>R b d=R N e u d = R, i h i mat phang (P) va mat cau S ( ; R) chi c6 m o t d i e m chung nhal N e u g o i H la d i e m chung ay thi H goi la l i e p d i e m cua mslt can vdi milt phiing Luc (P) se goi la liep d i c n v d i mat cau c N e u d < R t h i mat phang (P) vii mat cau S(0; R) cat theo mot giao luye'n la dirdng Iron Hinh chie'u H ciia O Iren (P) chinh la l a m cua difdng tron giao luye'n N e u g o i r la ban kinh cua dif5ng Iron giao luye'n Ihi: Neu d = |Ri - R2I > 0, k h i hai hinh cau tiep xiic v d i Boi (liCQiuf IISG Hinh hoc khon;/ (ji(in - J'haii IIiiij Khdi R rang M, N, A, D thuoc dU'cJng Iron giao luye'n Nhif the MADN noi rieng la mot lu" giac noi ticp Dicu mau Ihuan vCfi kcl luan Ircn Vay giii thicl phiin chiJug la sai Do la dpcm 'I'hi du 2: Cho hnih chop dinh S va day la mol da giac idi A^Aj A„ (n-giac loi) Tim dicu kien can vii du de ton tai hinh ciiu ngoai ticp hinh chop, ti'rc l;i dinh S ciia hinh chop va cac dinh A|, A , A „ ciia day dcu nam licn mol mat cau c Ne'u < d < |R| - R2I, hai hinh csui dyng Giai f Ncu d = 0, hai hinh cau ddng lam (khi O i = O2) lift) < ;-i + 1$ Ciic l)5ii loan chon loc ve hinh cau ,51 (BCM) n (SAD) = MN, d o N e SD va MN // BC (tufc MN // AD) Vi BA AD =^ SA AD (djnh li ba du'dng vuong goc) \ Vi the MADN la hinh thang vuong thirc sir (do MN < AD) Do MADN khong phiii la tu" giac noi liep Gia thiet phiin chiTng diem A, B, C, D, M, N cung nam tren mot mat cau (-rf )nao Khi mat phang (SAD) phai cat (if )theo mot giao tuyen la mot du'dng Iron DiOu kien can: Giii su" ton tai hinh cau Ulm O ngoai liO'p hinh chop S.A1A2 A,, tiJc la la CO OS = OA, = O A = = 0A„ (1) KcOHl(A,A2 A„) => HA, = H A = H A n (2) Dang thu-c (2) chlTng to day A,A2 A„ la mot da giac noi liep '2 Dieu kien dii: Giii siif A , A A n lii mot da giac noi liep Goi H la tiim u Si! 0'()i) III dU'cJng Iron ngoai liep day Qua H di/ng du'Cing thang A vuong goc vdi mat phang A1A2 A,, Ve m;lt phang trung Irifc (71) ciia mot canh bat ky ciia hinh chop (chang han SA,) Do A khong song song vdi (n) nen giii suf A n (n) = Khi la thay OA, =0A2 = = 0A„ OA, = O H Tijf suy O lii lam hinh cau ngoai liep hinh chop S A,A2 A„ NhiT vay, la CO kel luan sau: Dieu ki?n can va du de hinh chop S A,A2 A„noi liep mot hinh cilu lii: Da giac day A, A2 A„ la mot da giac noi liep Nhan xet: TiT thi du tren, ta nil cac kel luan sau: , , , , „, Moi hinh chop lam giac (tiJc la moi lii" dien), moi hinh chop deu lii hinh chop noi liep mot hinh cau Khi hinh chop da thoa miin dieu kien tren thi lam liinh c;tu ngoai ticp ciia no Xcic dinh iheo cac bu'dc sau: - Xac dinh tiim H Axil^ng Iron ngoai liep day A,A2 A„ - Qua H difng difcing thang A vuong goc vdi mat phang (AiA2 A„) Ve mat phiing irung inic (n) ciia mot canh ba't ky ciia hinh chop Giii siir A n (71) = Khi do, O la lam hinh cau ngoai liep cua hinh chop S.A,A2 A„candu'ng 173 T3di duTliui TTSG TTinlt hgcl^cHonij ainu I'lmn Iliiii A/uu T r o n g cac irifc^ng hdp sau day mat phang trung tri/c (TT) C the thay bang Ta C O SA, = yfsH^+A^H^ di/dng trung trifc a K h i hinh chop la chop deu ( v i k h i A qua dinh S cua hinh chop) s ,^ i.'ij ',,! f''it;;rv':M ;vr,:(! ii ;:;•:;);/( SK SO SH SA „ SK.SA, (6 day R la ban k i n h hinh cau can t i m ) R = SO = SH SA, ^.SA, NhiTvay K h i hinh chop c6 mot canh vuong goc R = v d i day (A|A2 A„) (giii su'do la SA|) Luc g o i {%) la mat phdng xac dinh b d i A 71 K h i O la tarn hinh cau ngoai tiep can tini A, K h i hinh chop c6 diTdng thang A qua S Luc ve trung triTc cua mot Chiiy: canh baft k y (giong nhu" tru'cfng hdp a) 7C N o i r i e n g neu n = , c p = - , t a c : R = ^ sinA,HM G p i H la t a m cua day B C D Trung triTc cua A B (ve tam giac A B H ) cat ^ 2sin- Trung irifc cua SA, c^t SH t a i O => O 1^ t a m hinh cau ngoai tiep cua hinh chop da cho CO SMH — • Giai cua hinh cau xac dinh nh\i sau: - M ^ = 71 tie'p tiJ dien R = A0 = = 71 5^^^ = A H tai O K h i O la tam hinh cau ngoai tiep ti? dien A B C D va ban kinh R A ; H A = — ^HA, ^ ' M i n h hoa 2: Cho tiJ d i e n deu A B C D canh bang T i m ban k i n h hinh cau ngoai I K e SH (A|A2 A„) k h i ta c6 g6c tao b d i mat ben va day nen S M H = Tir d6 suy a 7t S H = M H tan(p = M A j c o t A , H M tan(p = —cot— tancp • n 174 a ( l + cos^ ~ tan^ cp) n ^ 271 2sin — tancp , n ^ a d + cos^^an^-^-) S K e H M A , A2 => M A , = M A = ; 2sin~ t a n mat ben va day bang cp Hay xac dinh tarn va tinh ban k i n h hinh cau ngoai tiep hinh chop , ,t:vv| liljiv:"'''' - c o t - t a n cp n ^ X e t cac t h i du minh hoa sau day: M i n h hoa : Cho hinh chop n giac deu S.A1A2 A,, canh day bang a, goc cua Ta SH s i n - c o s - t a n (p n n Trong (71): d n A = O (1 + cos^ " tan^ cp) n s i n 2^ '7t' a(l + cos^~tan^(p) n R = " • Trong (71) ve du'(:fng thang trung trufc d cua SAi ^ cos — tan"cp + l n Hai tam giac S K O va SHA, dong dang nen ^ A„ c 'I" n sin io va SA, ( A / / S A , ) ' a 7^ — col —tan + .;,, ,f, b i = AK AB AB AB cosKAO 2cosBAH — "^^^ AB Ta CO A B = ; B H = o A H = | B M = | ^ = ^ T VAB2-BH^-^1-|=^ • Thay who (*) va co R = j-j^ = V6 — B (hcQnfj IISG Jlinh hoc khoncj gian - Phan Iluij Khdi Minh hoa 3: Cho li? dicn ABCD \(M AB = AC = a, BC = b Hai mat phang (BCD) va (ABC) viiong goc vc'li va mSc = 90" Xac dinh tarn va tinh ban kinh mat can ngoai lic'p li? dicn ABCD Ihco a, b Giiii Kc AH B C => AH (BCD) (do(ABC)l(BCD)) n Do BISC ^ 90" ncn H la tam du'cJng Iron ngoai licp tam giac BCD Ve trung trifc cua AB cat AH tai O, thi O la tam hinh cau ngoai tiep tiTdicMi ABCD va ban kinh R ciia hinh cau tinh nhii" sau: AB.AC.BC , R = AO = 4S ABC — (ap dung cong thiJc lu'cing giac quen bie't S = abc ) 4R AB'.BC AB' 4^ BC.AH 2AH Thi du Cho hinh chop S A A A , , Giii si? ton tai hinh cau noi tiep hinh chop, lu'c la hinh cau tiep xuc vdi ta't cii cac mat ben va day Goi r la ban 3V d day V, Spp Ian lu'c;! la the tich vii kinh hinh cau Chu"ng minh: r = ^— dien tich toan phan cua hinh chop Giiii Goi I la tam hinh cau npi tiep Khi cac hinh chop I S A A , ISA2A3, ISAn_|An, ISA„Ai va hinh chop IA|A2 A„ c6 chieu cao dcu bang r s ^'' Goi S|, S2, , S„ va S tU^Png iJug la dicn tich cac tam giac SA|A:, SA2A3, SA„A| va dien tich lam gidc day AiA2 A„ ^ Ta CO ^ - 'V1.SA1A2 + ^ I S A A + ' ^ I , S A n A i + "^[.SA]A2 An =>V= ir(S|+S2+ 3V + S„+S) = ir.S-rp => r = Nhdn xet: i Nhd cong thtfc noi tren cho phep ta xac djnh ban kinh r cua hinh cau np' tiep hinh chop ma khong can xac dinh tam hinh cau npi tiep Xet minh hpa sau day: Cty TNIITI MTVDWHKhnng Viet Cho ti? dien OABC, OA = a, OB = b, OC = c va OA OB, OA OC; BOC = 120" Tim ban kinh hinh cau npi tiep tu" dien f Ta CO OA (OBC), de thay AB = BC = a >/2 ; BC = a 73 ^ Ta CO V = VoABc=^SoBc-OA • ,onO a = ^ " ^ ^ = -1 -1a sinl20 32 12 Gpi M la trung diem cua BC thi: AM = VAB^-MB^ ABC = 15 V I the STP = SoAB + SoAc + SQBC + SABC a^ a^ Do vay r = 3V •+ - a^VTs a^ a 73 + 73+715 Nhcic lai mot ky niem: Day la bai toan tuyen sinh vao Dai hoc Tong hpp Ha Npi nam 1964 Tac gia cuon sach lam bai thi tren lijc thi vao khoa Toan cua trU'dng tai tru sd 19 Le Thanh Ton - Ha Npi Tac gia la cifu sinh vien cua tru'5ng Thi du 4: Cho hinh chop n giac deu SA1A2 A,,, canh day bang a, goc cua mat ben va day la Xac djnh tam va tinh ban kinh hinh cau npi tiep hinh chop Giai Ke SH vuong goc day A|A2 An Do hinh chop la deu nen mpi diem tren SH deu each deu tat ca cac mat ben Do do, tam I hinh can npi tiep hinh chop la diem nam tren SH cho I each deu mot mat ben nao (chang han ( S A A ) t \a day A|A2 An) Gpi M la trung diem cua A1A2, HM A , A =^ SM A , A (djnh li ba du'dng vuong goc) Ta CO SlvlH = (p vi no la goc cua ( S A A ) tao vdti day giiic SMH, ve phan giac cua SMH, no cii SH tai I A,A2 A„ Trong tam 177 BSi dudng MSG Hinh hgc khdng gtan - Phnn Huy Khdi Ke I K i SM D o ( S M H ) ± (SA.Az) Ctij TNHH I K (SAjAj) n Nhdn V i t r e n d a y A B C D x e p diTdc q u a c a u , ngi/cti ta c k h a n a n g d e x e p t h e m (1) q u a c a u d a t t r e n q u a c a u di?di ( t h e o l a p l u a n t r e n ) H M = MA2C0tMHA2 =-cot- n T i r ( ) , (2) suy r a : r = (2) n f i l l d y 6: T r o n g m a t p h d n g (P) c h o diTdng t r o n t a m O , b a n k i n h r v a A l a m o t -cot-tan2 n d i e m C O d j n h t r e n d i f d n g t r o n G o i A l a d i f d n g t h a n g v u o n g g o c vcfi ( P i t a i O c h i e u cUa B t r e n ( P ) G i a su" k h o a n g e a c h tiT B x u o n g ( P ) l a b T r o n g t h i d i i t r e n d a t r i n h b a y e a c h x a c d i n h t a m h i n h c a u n o i t i e p vdi m o t h i n h c h o p deu Q u a d6, ta c u n g thay v d i m o i hinh c h o p d e u , d e u t o n tai -y- 4,' : •== -iSi U • Giai C O t h e x e p v a o h o p 13 q u a b o n g b a n k i n h r k h o n g ? ( G i a suT c a c q u a b o n g G o i ( ) l a m a t p h a n g t r u n g trifc ciJa AB k h o n g b i e n dang x e p chdng l e n nhau) D o O B ' > a n c n A ;^ B ' =^ A B k h o n g s o n g s o n g v d i A, n c n (TT) Gial D' la t a m cua qua cau K h i => | co' d i n h R o r a n g m o i d i e m t r e n di/tfng t r o n ( O , r ) d e u e a c h d e u O i L a y B m a t p h a n g (O1O2O3O4) song s o n g v d i ^ ( A B C D ) va each no m o t khoang A -, N l a d i e m l i i y y t r e n d i f d n g t r o n T a co ^ 0,N O l a t i i m ciaa q u a c a u thu" d a t Vay t r e n q u a c a u n a y K h i d o c a c qua c a u O5.O1O2O3O4, O5H chinh G i a sur B ' O c a t d i / d n g t r o n ( O , r ) t a i A ' , A ( x e m h i n h v e ) A'A"B c a o cOa h i n h c h o p k h i N , A , B l a ba d i e m k h o n g thSng h i i n g cung n a m t r e n m a t c a u | , b a n V i h i n h c a u t a m | , b a n k i n h | B la h i n h c a u c o djnh => d p c m ma canh day va canh b e n deu O5H la dirdng = 0,A = 0,B d i r d n g i r o n n g o a i t i e p l a m g i a c A N B c u n g n a m t r e n m;lt c a u t r e n b a n g 2r Goi i k i n h | B V i m o i m a t p h ^ n g c i t t h i n h c a u t h e o g i a o t u y e n l a du'dng I r o n n c n d o i m o t tiep x u c ngoai v d i nhau, n e n O5.O1O2O3O4 l a h i n h c h o p ti? g i a c deu : theo nhiin x e t tren ' -7' - b ^ n g r Goi phai c;tt A v a g i i i siY: A n ( n ) = O i qua c a u dat k e d ve goc • ' b e n t r a i c u a h i n h h o p G o i O i , O2, O j , day T i m b a n k i n h c u a m S t c^u ( S ) t h e o a, b , r khan T h i d u 5: C h o h i n h h o p d i f n g d a y l a h i n h v u o n g b a n g r , c h i e u c a o l a 3,5r H o i O4 B i e t N la m o t d i e m d i d p n g t r e n diTdng t r o n ChiiTng m i n h r a n g cac dirdng i r o n n g o a i t i e p c a c l a m g i a c A B N la n a m t r e n m o t m a t c a u c o d j n h (S) V i e c t i m t a m h i n h c a u n o i t i e p cija m o t M n h c h o p ( d l n h i d n k h i n o t o n tai) noi chung la rat kho Xet la m a t p h a n g q u a t a m O i c i i a h i n h c a u ( S ) n o i d c a u 1, n e n d i r t t n g t r o n n g o a i tiO'p l a m g i a c B A ' A " c h i n h la difiJng t r o n I d n c u a m a t c a u ( S ) la k h o a n g e a c h t i f t a m q u a c a u thi? n a m Vay d e n m a t p h ^ n g (O1O2O3O4) R r a n g : t a m giac B A ' A " A p d u n g c o n g i h i t c : 05H= -OjH^ =yj(2rf +{Tyf2f = thi? d e n d a y ( A B C D ) c u a h i n h h o p l a : r%^ + r = r(2+V2 ) M H A j MTV DVVH = A'B.BA".A"A' 4S 5;, tiep , ,^/b^ + ( r + a)^ yjh^ + (a - r ) ^ 2r 4.- Q A•A".BB• 7b^+(r + a)^Vb'+(a-r)^ 2b 179 Doi fludng IISG Ilitih hoc kh6ng gian - Phan Iliiy Khdi T h i du 7: Trong mat phctng (P) ve nuTa diTdng tron dUdng k i n h A B = 2R T r e n A B Cty TNIIII MTV DVVH Khnng ViH '\U du 8: Cho A B C D la ti? dicn co cac cap canh doi doi mot bang (tiJc lay d i e m H Tij^ H kc duTcJng vuong goc vc^i A B cat mJa difcfng tron trcn la, A B C D la ti? dien gan deu) ChiJng minh tang lam hinh cau noi va ngoai tiep M G o i I la trung d i e m cua H M Nu"a difdng thang vuong goc v d i (P) tai I cfii cua no trung mat cau diTtJng kinh A B tai K I^?,f:2 ' ChuTng minh dai liTcJng K A ^ + KB^ khong phu thuoc vao v i I r i cua H • Giiii ' ncn A K B = 90" Trong tam giac vuong V a y H , K tu^dng xing la tam cac du'dng Iron A K B theo dinh h Pitago, ta co ngoai tiep cac tam giac A B C , B C D fito AmA u K A ^ + K B ^ = A B ^ = 4R^ = const TCr gia thict suy A A B C = A B C D (c.c.c) Do ladpcm ncn cac ban kinh du'dng Iron ngoai tiep Do I H A B => K H A B (dinh l i ba B D cua hai tam giac bang => H A = B K dir5ng vuong goc) => K H I = a chinh la M a t khiic, O A = OB ncn hai tam giac Q goc tao bcfi hai mat phang ( K A B ) va (P) * 54B i n O A H va O B K bang => O H = O K ^ ' til' • ; Trong cac tam giac vuong K A B va M A B , V a y O each deu mat ben ( A B C ) va day ( B C D ) ta CO K r f = H A H B ; M t f = H A H B Tu-dng tir, O cung each deu bon mat cua tu" d i c n A B C D , tiJc lii O la tam hlnh Ttrdo s u y r a K H = M H cau noi tiep A B C D V i the hinh ti? dien gan deu A B C D co cac l a m hlnh cau ^ i777 HI D o cos K H I = cos a = — HM =— KH 2KH noi va ngoai tiep trilng => dpcm ssiJKn '• V Thi du 9: Cho bon hinh cau ban kinh r lifng doi mot tie'p xiic v d i Hlnh cau i thi? nam tie'p xuc v d i ca bon hlnh cau tren T i m ban kinh hlnh cau a = 60" = const => dpcm Giai X e l t u ' d i e n A B K I T a m | cua hinh cau ngoai tie'p ti? dien thoa man dieukicnO,A = 0|B = 0,K = 0,l/^- * ' N6iriengtac6 0|A = 0|B = 0,K '^ (/ V a y tam O i nSm tren di/c(ng vuong goc v d i ( A K B ) tai tam di/dng tron ngoai '•' tiep tam giac A K B Do A K B = 90" nen tam dtf^ng tron ngoai tiep tam gia*-" A K B chinh la trung d i e m O cua A B Chii y r i n g O la d i e m co dinh M a t ph^ng ( K A B ) cat (?) theo giao tuyen A B co dinh v i tao v d i (P) mot goc khong doi = 60" (theo cau 2) => ( K A B ) la mat phang co dinh Du'dng thang Ox vuong goc vdfi ( K A B ) co dinh tai d i e m O co dinh, nen Ox CO dinh T a m O, hinh cau ngoai tiep tu" dien A B K I nam tren O x D o la dpcm Goi | , O2, O,, O4 la tam ciia hlnh cau ban kinh r K h i , ia lu' d i c n deu Ccinh bang 2r Goi O la lam ciia hlnh cau thu" nam tie'p xiic v d i ca hlnh cau tren va gia stir x la ban kinh cija no f v i .r( X e l hai tru'dng hdp sau: H l n h cau thu' nam tie'p xuc ngoiii fp^tnn., jnY);) X) linif- v d i hlnh cau tren K h i la co: i 0 , = 0 = 003 = 004 = r + x Q, / ^I (1) Dang thiJc (1) chiJug id O cung la tam hlnh cau ngoai tie'p ti? dien 0,020,304 \ 04 ^ i K e O H (O2O3O4) Theo thi du 2, < O lii giao d i e m ciia trung triTc , vdi , H H M BSi dudng HSG Hinh hoc kh6ng gian - Phan Iluy Khni Ta c6: R = , 0,0 O.K = cosKOjO _ OA 20,H 0,0 Tir (1) suy de IJ la du'cJng vuong goc 2—^0,02 chung cua BC va A D , ta can c6 IJ A B 2cos020,H 4r^ Do J la trang diem cua A D nen IJ A B , o l A = I D A A B C = A D B C o b = c ' (2) V a y b = c la dieu k i e n can t i m 2^0,02^-02H^ Do O2H = - O M 2r^y3 (d day M \k trung diem cua O3O4) = - • 3 nen tijf (2) suy R = 2rV3 Do b = c nen tam giac can C A D (dinh C), ta C O CJ A D K e t hdp v d i D I BC suy I va J nhin r^/6 C D du'di mot goc vuong => hinh cau dU"cVng kinh C D qua I va J rV6 Laitheo(l)tac6R = r + x 'dC 11 /'I'', • X , = r Gia O O , = 0 = 00.1 = 0 = x - r d day theo phan thi R = rV6 - • X = + r 76 Khib = +1 goc phang cua nhi dien canh BC cua tuT dien (tiJc la goc giffa hai mat phang V d i dieu k i e n nao cua b, c thi diTdng thang noi I , J la diTdng vuong goc Chung cua BC va A D ( d day I va J tu-dng ufng l i i cac trung d i e m cua Giasufb = c = a^/3 • Xac dinh a de hinh cau di/dng kinh IJ tiep xuc vcli CD Giai c = ^ : o A I ' > =:Di= VAB^-BI^ = jb^-i^ = 3a^ a" aV2 IJ B C =^ C I = CF = I (do C I va CF la hai tiep tuyen cung xuat phat liT C doi vdi hinh cau dutfng kinh IJ) Tir D F = C D - CF = , i ^ DJ = DF = -(73 -1) '•-^ BC A D ) ChiJng minh rang k h i ay hinh cau diTcfng k i n h C D qua I va J thay Gia suT hinh cau 6\ibng kinh IJ tiep xuc voti C D tai F (chu y theo cau 1, thi T h i d u 10: Cho ti? dien A B C D c6 BC = a, A B = A C = b, B D = D C = c G o i a la • De khoang each tuT trung d i e m O cua IJ xuongDCbang ^IJ V a y triTcfng hdp nay, bin k i n h hinh cau thi? nam la: r j = r aV3 dirdng k i n h IJ tiep xuc C D k h i va chi (3) O1O2O3O4 va ta c6: f i x - r = R ! = G o i O la trung d i e m cua IJ Hinh ctlu Dang thiJc (3) chiJng to rang O cung la tam hinh cau ngoai tiep tuT dien A B C va D B C ) c BC, tir A l b = a H i n h cau thu* nam tiep xuc va chiJa hinh cau bang d i = A I D c h i n h la goc phang nhi dien canh K h i do, theo quan he ve s\i tiep xiic giffa hai hinh cau, ta c6: > b Trong triTcfng hdp nay, ban k i n h cua hinh cau thoa man la: r , = r.( — - ) sur a ~ J JD D ^2 ( ^ - ) T a c o sin — = s i n J I D = — = ID a72 => cos a = I-2sin^ Y 73-1 ^ / ^ 72 273 - =:i a = arcos( 273 - ) Goi I va J tu'dng iJng cdc trung d i e m cua BC, A D Ta CO A B = A C => A I BC, TCr BC l ( A I D ) => BC I I J D B = DC => D I l BC (1) d u 11: Cho ti? dien A1A2A3A4 Goi h i , hj, h j , h4 Ian liTdt la bon chieu cao ciia ttf d i O n ke tfif A , , A2, A3, A4 Goi r la ban kinh hinh cau noi tiep tu" d i O n BSi ductng HSG mnh hoc kh6ng gian-Phanlhiy ChuTng m i n h - = — + — + — + — r h| hj ChuTng minh h i + h2 + h , + h4 > 16r Cty TNHH Khni Giai J hinh cau n p i l i c p nghiem lai dieu nay) Syp S| "f" S2 "f" - 2- (1) S-^ "4" (2) r 1-2^='-^ (3) r , h3 (I) d d a y , , S3, S4 liTdng iJng l a d i e n l i c h c a c m a t A A A , A A A , A1A2A4 1-21 r Tir(l)suyra ^ OSi S-i 0-) St -—^+ — 3V + — 3V + 3V 3V — (2) — +— +— + — '4; Mat khac ta CO V = i S | h | = ^ S h ^ ^ S ^ h , = ^ S h - ^ - - ( i 3V hi 1.4) (3) (4) Cong tirng ve (1), (2), (3), (4) va co /- \ 1 1 _ r, + r; + r^ + r4 4-2r Q to Theo thi du 11, la co - = ^ + ^ + ^ + - r h3 h, h, ,11 • A, r h, Lap luan tu'dng t i f , la co 3V h ' l = h | - 2r • A2 i h i du ta co: 3V • De t h a y , h a i tuTdien A1B2B3B4 va A1A2A3A4 la h a i t i i r d i e n d o n g d a n g n e n la co: G o i r la ban k i n h hinh cau Theo r = 4r'^,f dpcm Thay (3) vao (2) suy r — +— +— +— h, h-, h , h^ dpcm A p dung ba't dang thtfc C o - s i , la co: (ri+ r2 + r3 + A p dung ba't dang ihiJc C o - s i , ta c6 (h|+h2 + h3 + h4) iijl 1 1 >16 (4) — I +" — + h3— + 114— Tir c a u va (4) suy h|+ h j + h3 + h4 > 16 r — r, + — r2 + — r3 + — '4; A A A , A A A , A A A va A A A G o i c a c lurot la B2B3B4, B B B B B B , B1B2B3 X e t liir d i e n liep d i e n la" nho A1B2B3B1 v6i Giii sijf O A = a, OB = b, OC = c D a l S = SABC- rhng: r = a + b+c Giai V i O A O B , O A O C => O A (OBC) nen tiTOng u^ng n o i l i e p bo'n ti? d i e n noi I r e n ChiJng m i n h rang r i + r2 + r3 + r4 = 2r ChiJug m i n h r a n g — + — + — + — '/ ' ' V = V0.ABC = VA.OBC = -SoBC-OA = abc S , = S O A B , S2 = S Q B C , S3 = SOAC, G o i r la ban kinh hinh cau n o i liep li? dien O A B C ChiJug minh S, + S2 + S3 - S G o i r,, r j , r , r4 la b a n k i n h c u a h i n h c a i ' (6) > 16 Thi d u 13: Cho tiir dien vuong O A B C dinh O, life OA, O B , OC doi mot vuong goc d i e n v d i h i n h c a u n o i l i c p n a y , c h o m o i l i e p d i e n l i f d n g ifng s o n g song A2B1B3B4 A3B1B2B4 A4B,B2B3 'Da'u bang xay r, = r2 = r = r4 A , A A A la itf d i e n deu T h i d u 12: Cho luT d i c n A , A A A V c h i n h c a u n o i lic'p tu" d i e n Sau d o v c licp mat T i r p h a n va (6) suy — + — + — + — > - => dpcm j ,,tirii ankh ihi (y^ Dau b a n g x a y h| = 112 = h3 = h4 A A A A la liJ d i e n d e u v(3i c a c (1) - ibc lidi diC&iic) IISCz ITinh (jian -Plum hoc khcmg Hull Cty rmiH Khdil A p dung cong thiJc thi du , ta c r = V _ Syp 3V , .A:Aabc S, + S2 + S3 + S d, a, 33 DSu bang xay - — - — b b , b , : "2 "7, (2) 2(S, + S2 + S3 + S) abc(S| +S2 + S - S ) _ abc(S| + Sj + S3 - S) (S, +S2 + S ) ' - S M 2(2S,S2 +2S,S3 +2S2S3) : i = abc (a + b + c) Thay (5) vao (4) va c6 r = ^^+^2+S^-S (4) ^ ; a PA, = - 1 1 ^^^^ h^ • d u l l , t a c - +- +- +r h a b c • JSJU^JDI ( S B C D - P A , + S A C D - P B , + '''ABD P C ] + S A B C - P D I ) ^ (SBCD + S A C D + SABD + SABC)^ (l>s 3V 'TP T > — => la dpcm r I863 3V 3V; 3V3 a+b+c (2) r athiJcb C oc- s i , ata+ c6 b+c va theo bat dang 1- > 3 (3) V a n ihco bat dang thiJc C o - s i , ta c6: (a + b + c)^ > QA/JIVC^ B | Tilf (3), H| Da'u bang xay o W O A = O B = OC va O A , O B , OC d o i mot vuong goc v d i nhau) (4) (4) suy (2) dung => dpcm a = b = c O A B C la ti? dien vuong can dinh O (tiJc la Thi du 16: Cho tiJ d i e n vuong O A B C dinh O (tdrc la O A , O B , OC d o i mot vuong y rkng SBCD-PA, + S^CD-PB, + SABD-PC, + SABC-PD, = V „7 h Va^b^c^ Js^^^Jc^; h, = 1^ 3V3 ->-+-+-+ V d day V = VABCD, ngoai ve phai cua (1) chinh la S\ !; 1 1 ; 'd4 — iBCD_ ^ ^ACD ^ f A B D ^ ^ ^ B C PA, PB, PC, PD, A p dung c n g l h i ? c r = c^ Tif s u y ra' Theo bat d i n g thiJc Bunhiacopski, ta CO Til thay vao (1) c6 3TY $: S' =^ T > b^ lu-dt tijf O A B , C la h , a, b, c Theo thi ^ TiJT (1) V PC, b, = V ^ B C D - P A , ; b2 = V S A C D - P B , ; b3 = DC a^ TiJ dien O A B C co chieu cao ke Ian 'ABD \l PB, B^x/s Theo ke't qua cd ban (xem ChiTdng 2) la c6 + ^ ^ B D I ^ABC > S PC,, PD, ~ r 'ACD I K e O H i ( A B C ) va dat O H = h Giai D4ta,= ,' Giai va r tifdng iJug la dien tich loan phan vii ban kinh hinh cau n o i lie'p ti? dien PB, j; 1 1 D , Ian liTdt la hinh chieu cua P tren ciic mslt B C D , A C D , A B D , A B C Goi S + • cau n o i tiep tu" d i e n Chtfng m i n h - > — + — + - + -r a" , , , b " a+b+c T h i dy 14: Cho tu" dien A B C D , P la d i e m y ben tiJ dien G o i A , , B,, C,, PA, "4 vuong goc v d i nhau) Gia suf O A = a, OB = b, OC = c G o i r la ban k i n h hinh a +b+c Chii-ng m i n h rang: T = b Xhi du 15: Cho O A B C \k tuf dien vuong dinh O (life la OA, O B , OC doi m o t ' (5) ^ Viet d day I la l a m hinh cau noi tiep liJ d i e n A B C D Ta CO 4S,S2 + S , S +4S2S3= (ab.bc) + (ab.ca) + (bc.ca) J Khang a4_ PA, = PA2 = PA3 = PA4 O P = I A p dung dinh l i P i - l a - g o met rong khong gian, ta c6: S,' + S2' + S3' = S2 (3) abc(S| + S2 + S3 - S)' Tilf (2), ( ) ta CO r = — 2(S, + S2 + S3 + S)(S, + S2 + S3 - S) - MTV DVVH ' ll goc v d i nhau) G o i R, r, h , V W d n g uTng la ban kinh hinh cau ngoai tiep \\i B d i e n , ban kinh hinh cau noi tiep luT dien, chieu cao cua ti? dien ke tiJf O va P, the tich cua tu" dien ChiJng minh rang: R^rh nen liT (2) suy R^3 " + 3>^ h i ^ > l + 73 r Bdi dudtig HSG ITinh hoc kh6ng ijian - Phan Ihiy CU) TNHIl Khdi Giai D a t O A = a, OB = b, OC = c Ke OH (ABC), ihi OH = h 3V Theo T h i d u 3, ta c6 r = « - ' ^ d day STP va SABC Ian liTcft la dien tich \ 'H 3V Theo thi du l a i c6 r= 3V V(h-r) STP-SABC R^9V2 R^rh d day SXQ - SQAB + SQBC + SOCA Vayltr(l)c6 Xi^ SXQ 3R^ -'i (1) ab + be + ca 3V STP (2) +^ +^ +S ABC Va l a i 1 ,_L^orf= 2.2 a^b^+aV+bV _ b ^ b^+e^ = > SABC = - B C A H = - V b + c 2 A OA^ abc Ke O H B C => A H B C (dinh l i ba => A r f = a^ + trung d i e m cua B C vii M x vuong goc vc'Ji (OBC) tai M K h i de tha'y: IVb^+c^+a^ Tur (2) o ab + be + ca < a^ + b^ + c' (a-b)^ + (b-c)^ + (c-a)^ > Vi (3) diing suy dpcm A, (3) umiii O A B C la tt? dien vuong can dinh O 1/ r i (4) • A p dung bat d^ng thiJc Bunhia copski, thi: - + Va M a t khac van theo t h i du 15, ta c6: 1X8 + •Of! M (9) abc ab + be + ca + \/a^b^ + a^c^ + b^c^ 4& ,,„, (lU) Va^+b2+c^>MVc2=V3^, T h e o t h i d u 15, t a c o : - = - + - + - + r h a b c •;, + c^ , ^, R + b^ + c^(ab + be + ca + Va^b^ + a^c^ + b^c^) tu'do suy ra: — = r 2abc A p dung lien tiep bat dang thi^c C o - s i , ta c6 D a u bang xay o a =b = c , =:lVaV+aV+bV Thay (9) vao (8) va c6 r = o dpcm (8) ^ dUdng thang M x vii di/tlng trung triTc cua O A (ve ( A O M ) ) , d day M la = h + l) hay - < + V3 ' r BC = V b ^ T ? , A r f = OA^ + O r f ^'^^^^/^^ < ^ o ab + be + ca V ^ ' » e ^ = ^ / ^ £ W Tird6theo(10).suyra: R = - ^ ^ ( ^ / ^ ( ^ ^ / ^ ) ) _ 3^-373 la dpcm 2abc r Da'u bang xay o a = b = c 189 lidi diCOng IJSG Hinh hoc khony (jinn - Phan Cong tCrng ve (1), (2), (3) co: 2 b^+a^-c^ M w w ^ |(b' + a^ - + y^ + Khdi =^ ^ ' l b^+a^-c^ Cty 42{h^ + a^ - )(b^ - a ' )(a^ + UKC^;:,, - b^) V = -abcVcosA.cosB.cosC - a^ )(a2 + - b^) i n (a x 'i cua hinh hop chi? nhat A E C F G B H D noi tren G o i R la ban kinh hinh cau ay, , — ; — ; — =.-V2(a2+b^+c^) 'i i ' ' ' , ' ' f ^ w i ,u^h,/-, '.?,v V 3) Ta CO I | = ^ A p dung djnh l y ham so cosin tam giac K H O | ( d day K J l l , K = H K ' + H O , - 2HK.HOiCosa hay a^ a^ a^ =— + — - y c o s a hi Cong tijfng ve ta c6 a^cosa + b^cosp - c^cosy = ^^ T h i du 7: Cho M la m o t d i e m bat k y nam m o t tiJ dien gan deu A B C D M M , , M M , MM3, M M Ian liTcft vuong goc vdi cac mat B C D , C D A , D A B va A B C G o i r va p Ian li/dt \h ban kinh cija mat cau noi tiep va ban kinh diTdng tron ngoai tiep moi mat cua tiirdien n^y Goi A, B, C la ba goc cua tam giac ABC 1) ChuTng minh: M M ^ + MM2 + M M ^ + MM4 > 4p^ cos A c o s B c o s C , , •( i i hi • ?\ T h a y ( ) , ( ) v a o ( l ) c Y M M J = h = 4pVcosA.cosB.cosC (4) ip dung bat dang thtfc Bunhiacopxki, ta c6 1^ = ( M M , + M M + M M + MM4)^ < ( M M f + M M ^ + M M ^ + M M ) (4), (5) suy ra: J ^ M M ? > p ^ cosA.cosB.cosC L y luan tUdng tuT suy dpcm 1,;, (3) , • ui>uft,;: dpcm (5) i , ;l!T! iif>V \IQ >au bang xay ra: o M M , = M M = MM3 = MM4 Tif CO a^cosa = b^ - c l L y luan tiTcJng tir, c6 b^cosp = a^ - c^cosa = a^ - b^ 2) Chufng minh: - < — p , (2) Ta CO S = - b c s i n A = 4p p H Ian lifdt la trung d i e m cua D B va A B ) ta c6 I Viet :0a tam giac A B C ) m o i doan thang TiT suy tam hinh cau ngoai tie'p tu" dien chinh la tam VI;a Khang (ip dung dinh l y h a m so cosin tam giac A B C , d day A , B , C la cac goc )(b2 + cac cap canh d o i dien dong quy tai mot d i e m , va d i e m la trung diem ciia /b^ DWH => V = — V2(b^ + c^ - a^ )(a2 + c^ - b^ )(a^ + b^ - c^) 2) Ta biet rang mot ti? dien bat k i doan thang noi cac trung d i e m cua th!: R = IC = MTV DiTng hinh hop ngoai tiep tu" dien => day la hinh hop chi? nhat 4- TNIIH d day V va S tU'cJng iJng la the tich va dien tich loan phan cua tu" dien - vay suy b^+c^-b^ NhirvayV = - J hay V = ITiiy \> M la tam hinh cau noi tiep tu" dien A B C D ,„, ^, 2fAp dung l a p luan phan 1, ta thay k h i lay M = I ( I l a tam mat cau n o i tidp), ta CO 4r^ = 4p^cosA.cosB.cosC hay r = pVcosA.cosB.cosC = VcosA.cosB.cosC (*) P Theo ket qua lifdng giac, ta c6 m o i tam giac (khong c6 g6c tii) ^l ;::r^cosA + cosB + cosC „ ^ ' I ' • - nen ncu ke D H M N thi D H ( A B C ) TO (1) ta CO1t a n V +1 tan^p 1+ tan'y = 12 o HD^ = 12 •+ +VMH' N H ' PH' ' V i ( A B C ) ( D M N ) va ( A B C ) n (DMN) = M N , P? Khang VUH fhi du 9: A B C D la tiJ dien deu canh b^ng M va N la cac diem di dong tren Giai ^ , HD „ HD HP laco:tana = ; tanp = ;tana = MH NH PH Goi a 1^ canh cua tiJ dien deu: MTV DWH 1! PH^ a^ cua tam giac deu ''" ' A B C , vi the' MAH = N A H - " • • • c •} • • • c, T a c d S AMN = - ^ x y s i n M A N - ^ x y (2) • De thay M , N , P thang hang (vi no ciing n^m tren giao tuyen cija mat phang qua (1) M a t k h a c S A M N = ^ x A H s i n " + ^ y A H s i n " = ^ ^ ( x + y) (2) D H vdi mat phang ( A B C ) ) H M , H N , HP hip vdi cac dufdng vuong dtfdng vuong goc deu cd dai HM' = HN' = HP' = S =^ H M ' , H N ' , H P ' doi mot lap vdi Vay: HM^ •+ ^+H N ^ HP^ +y2/l +^ + :/^7(x + y ) - x y Vmin va S„,i„ o ^ Ro riing < x < l ; < y < l xy n/iii] }^JA O : J;) oli 3) TCr cau 2) suy Vmax va S^ax xy max HP = - ^ ^ 6cos(p =— 6cosA De thay ^ + cp = 120"; H + X = 60" =^ |a = 60" - ?i; (p = 60" + X''*^ i Theo cau 1) suy S = VSxy + — x y ( x y - l ) a73 MM HN = - ^ ; 6cos|a ^ Goi S la dien tich loan phan cua tu" dien D A M N , ta cd: ,ngoai gdc 120" Tir ta cd H M = • |) T a CO ngay: V D A M N = D H S A M N = • ^ • - x y — ' g d c ke tur H xuong B C , A C , A B C a c 7^ i T u r ( l ) v a ( ) s u y r a x + y = 3xy Gia suf A,, (i, (p la cac gdc lap bdi rcos^ -k + cos' (60" - A) + cos' (60" + ?.) xy ^n'ri Jojn.euj rrtfll 'Dat xy = t Theo cau 1) ta cd x + y = 3t (3) ioO m »^ - ^ ' ^ ^ ' V a y X va y la nghiem cua phu^dng trinh: f(z) = z ' - 3tz + t = (3) vi '' ' NhiJug gia trj cd the cua t la nhffng gia tri lam cho (3) cd nghicm Z i , zj De dang tinh diTdc cos^ X + cos' (60" - ?^) + cos^(60" + Tir (3), (4) suy (2) dung => dpcm =| (4) thoa man < Z i < Z2 ^ • Dieu xay va chi khi: , 261 Boi dudiig TISG Hhih line khonij (jUiii - I'lian Ihty Khdi - ( Z j +Z2) + 1>0 A>0 A>0 (z,-1)(Z2-1)>0 Z,/2 Z, + Z < >0 DVVIl Kluuuj ViH Giai Goi R \k giao diem cua OM vdi mat ,,r Z| - Z < Z|Z2 Z| +Z2 Ctij TNIIHMTV phing K L N Vice khang dinh R la >0 tam cua tam giac K L N tifdng difdng vdi >0 Z| + Z Taco A = t ^ - t ; Z | + Z = t ; z i Z = l khing dinh rang ba ti? dien MKLO, " A < MLNO, MNKO co cung the tich Gpi x, , , \\[\ y, z la khoang each tilf M tdi cac canh 9t^-4l>0 Tir he (*) tiTdng diTcfng vdi he , t-3l + l>0 '\ 3l_/3 + V2) M s A , " '•' '• NsC turn "»' Thi du 10: Gpi O la tam cua mot hinh tiJ dicn dcu TiT mot diem M bat ki ticn mot mat ciia tu" dien, cac du'cJng vuong goc tdi ba mat phang lai, vil' K, L va N la chan cua ba du^dng vuong goc Chifng minh rang difcfn^ thing OM qua tam cua tam giac K L N :",ij,fi if' 262 • i 27 Cac tiJ dien MLNO va MNKO cung cd the tich nhiT the TCf suy dpcm x=l ' )) Boi (ha'liuj nSCi Hinh hoc khong C h u y e n de CAC gian - Phnn IIiuj BAI T O A N QUI Ctg TNIIII Khni MTV b G i a i m o t b a i t o a n diTng h i n h , thiTc cha't la diTng d i e m N TICH T c C h u ' n g m i n h d i e m N diTng diTdc t h o a m a n m o i y e u c a u d e TRONG HINH HOC KHONG GIAN S a u k h i h o a n t h i i n h b a p h a n : t h u a n , g i d i h a n v a d a o , ta ke't l u a n v e C u n g nhiT c a c b a i l o a n t i m q u y t i c h ( h a y t i m t a p h d p d e m ) t r o n g h i n h hoc t i c h c a n t i m u iim\r, gnca i-.n '^sitjuri ^ p h i n g , b a i t o a n t i m q u y t i c h t r o n g h i n h h o c k h o n g g i a n t h u o c I d p cac b a i t o a n X h i d u 1: C h o diTdng Iron d u ' d n g k i n h A co' d i n h = R v a C la d i e m c h a y t r e n k h o D c g i a i b a i t o a n n a y c h u n g ta p h i i i t i c n h a n h q u a ba bu'dc: p h a n t h u a n , d t f d n g t r o n T r e n du'dng t h a n g A x d i q u a A v a v u o n g g d c v d i m a t p h a n g c i i a g i d i h a n q u y tich va p h a n diio d i r d n g t r o n lay d i e m S co d j n h Tit A k e A I T r t f d c he't c h u n g ta h a y d c c a p dc'n n g u y e n l y c h u n g c i i a v i c e t i m p h a n • thuan, g i d i han va p h a n dao cua quy tich G i a su" M la g i a o d i e m i> • r v " f s v VtYA (d M i e n D n a y p h a i g a n sat \6i ""^"T::, f{ n a y c h u n g ta p h a i c h i m i e n D m a d i e m c a n t i m q u y t i c h n a m t r e n d o ] Phan q u y t i c h c a n t i m ( I h i d i i n e u n h i f q u y t i c h la thuan: , Ta cd B C AC (gdc nufa d u ' d n g t r o n ) m a S A diTdc m i e n D n a y ngu'cfi ta thu"iJng silr d u n g c i i c bu'dc sau d a y : => S A (SBC) g i ? M u o n the' ta h a y v e thu" v a i d i e m c a n t i m q u y t i c h , r d i p h o n g d o a n m i e n D C a n l u l l y v d i cac b a n r a n g d i e u n a y do'i v6i bcin q u y tich k h o n g gian M a t khac A M chiJa q u y t i c h c a n l i m D a y la p h a n c o t l o i c i i a p h a n t h u a n ( v a c u n g l a c i i a t a p c o n D | c i i a D m a D | c h i n h la l o a n ho d a c b i e t la cac d i e m cifc b i e n (nhuT d a u m i i t c u a bien thien di dong) tai C doan r.,-, =^ S A AM e ( A I K ) , nC-n tir (1) cd S B AM ma A M ( A B C ) • li A M (SAB) \\'f,vJi-(Wir' => A M AB, tuye'n A x c u a du'dng Iron du'dng k i n h A B v e t r o n g (P) ~ JH C : : umhi ;p.tm ' s / K i t;^ •tJi-A'^im^ '.UA^ nu, = v;r-,,^ dao t r e n d u ' d n g Hon c h o n e u g o i 1', K', la cac h i n h chie'u c i i a A l e n S B , Trong • n B C thi M'B tai C Trong ( M I ' B ) , thi M T c a t du'dng Iron Ke AI' SB n SC = K ' T a se chu'ng m i n h r a n g A K ' SC V l A y A B , A x S A => A y phu-dng hu-dng c a n g i a i q u y e t N h i n c h u n g lu-dc d o c i i a n o nhU" sau: G i a suT M (SAB) ^ A y l S B L a i c d S B l A F e D | la d i e m n a m t r e n q u y t i c h , ta p h a i chu'ng m i n h r a n g t o n t a i d i e m N b i e n t h i e n t h o a m a n m o i y e u c a u c i i a d e b a i , m i l tiT d i e m N d o x a c d i n h c h o ta (P), dirdng kinh A B a P h a t b i c u b a i t o a n d a o c a n l a m Bu"dc n a y q u a n t r o n g d c h o c h i r o d u n g d i e m M e D2 (1) SCthlM' = rK' ' '' q u y t i c h C d t h e h i n h d u n g du'dc p h a n d i i o c i i a q u y t i c h t i e n h a n h t h e o cac day: A K I S B D a o lai lay m o t d i e m M ' t u y y t r e n tic'p t u y e n Ciy T a p h i i i chu'ng m i n h t o n 3) P h a n d a o c u a q u y t i c h c d t h e x e m nhu" la p h a n k h d nha't t r o n g q u a t r l n h t l n i sau hgnti Phan quy tich bien thien d diiu bai c h u y e n d o n g t r e n d o , va d a c b i e l l i i cac d i e m b i e n t h i e n d d a u b a i c h u y e n bu^c Gi(?i SC, D e tha'y k h i C c h a y t r e n d i f d n g t r o n , t h i M c h a y t r e n l o a n t i e p tuye'n n d i t r e n 2) P h a n g i d i h a n c u a q u y t i c h c h i n h l a p h a n c h i c h i n h x a c q u y t i c h , tu'c la t i m thfing t r e n d o diem (SAC) tu'c la M n a m t r e n Ucp bai loan t i m quy tich) d o n g t r e n do, va (SAC) (AIK) V I SA (ABC) DiTa v a o d i e u k i e n d a u b a i , chiJng m i n h r a n g D c d k h a n a n g la m i e n Stranl noi tiep chan L a i t h a y A I S B ( g t ) => S B da b i e t v e h i n h t r o n g h i n h h o c k h o n g *' ^'••} (SAC) vay A K l ( S B C ) : ^ g i a n C O t i n h cha't tU"cfng d o i c u a n o - lujo'Aiif D o ( S B C ) n ( S A C ) = SC m a A K thu'dng l a k h o h d n v d i q u y t i c h p h a n g L y d o v l v i c e v e h i n h v d i h i n h hoc p h a n g l a c h i n h x a c , c o n nhu" cac B C => B C ?• Giiii ; d o a n t h a n g A B , t h i m i e n D c h a n g h a n la di/dng t h a n g chiJa AB ) D e t i m - T i e n d o a n q u y t i c h , tu'c la p h a i h i n h d u n g du'dc D c k h a n a n g la m i e n S B , A K SC ( I e S B , K G S C ) cua B C va I K T i m quy tich d i e m M k h i C chay tren dufdng i r o n 1) P h a n t h u a n c i i a q u y t i c h la p h a n t r o n g t a m c i i a b a i l o a n q u y t i c h T r o n g p h a n ; uyfU? quy :=>SBl(ArM')=>SBlAK' i => B C (SAC) => B C l A K ' (*) (**) 265 Boi dUoiifj IISCi Ilinh hoc khong gian - I'han liny TO (*), (**) suy A K ' ( S B C ) Ctij TNIIII Khdi A K ' _L S C => dpcm Viet V a y quy lich ciia M (hay tap hdp cac d i e m M ) la tie'p tuyen v d i diTctng ki'nh A x Dirdng thang qua M vuong goc v d i ( M B C ) c;tt (P) tai R, diTcing thting qua A B tai A ve mat phang (P) M vuong goc v d i ( M C D ) ciit (P) tai S Goi I la trung d i e m cua RS T i m quy x-i\J :Vi,\':, tich I k h i M chay tren A x hai d i e m chuyen dong tUdng iJng tren a, b cho du-dng thang M N hOp V('(i Giai a, b nhffng goc bang T i m quy tich trung d i e m P cua M N /) Phan thuqn: Giai ' Goi A B la di/dng vuong goc a, A B ± b ^ - ' Ta chung cua a, b ( A e a, B e b) NhiT vay A B i:;./:,,Vw.iy:r.:y.r:.:,,^Jt, Lai d6 tha'y nam tren A D keo dai ve phia A ,t < , V i A D = A B => AS = A R Do I la trung d i e m cua SR, nen I nam tren n f e i dirdng thang A x ' la phan giac cua goc doi dinh v d i goc B A D (goc A ciia hinh vuong A B C D ) PC = PD Goi C va D ' lifting iJng la hinh chieu cua C va D tren ( T I ) R O rang C ' G 2) Phan duo a', D a o l a i gia suf I ' la d i e m y nam D'eb' tren nufa difcJng thang Ta C O a (PCC) (do a C C ) => a P C =^ a' P C TiTdng tiT ta c6 PD' b' S' la giao cua dtfdng th^ng qua M ' sau d a y : T r o n g (n) hay t i m vuong goc v d i ( M ' C D ) v d i (P); thi I ' la quy tich nhi?ng d i e m P each deu trung d i e m ciia hai during thang a', b ' Theo quy a', b ' ( T I ) R'S' Qua r diTng du'dng v u o n g goc v d i tich cd ban h i n h hoc phang ta c^t cho M ' va vuong goc v d i ( M ' B C ) v d i (P), gian qui ve b a i toan quy tich phang tao b d i hai diTcIng thang phai neu goi R' la giao cua du"cfng thang qua l o a n quy tich k h o n g la du"t(ng phan giac Ox va Oy cua A x ' Ta chifng m i n h ton tai M ' e A x V i the tijf PC = PD =:> P C = P D ' c6: Quy tich trung d i e m P cua M N 1(11 Trong tam giac vuong R M B , thi M A ^ = A R A B K e PC a, PD b ( C e a, D b) Theo gia thiet ta c6 A M N = B N M T L T suy hai tam giac vuong M C P ca D N P bang M R ( M B C ) , d day R nam tren Trong tam giac vuong S M D , i h i M A ^ = A S A D - •• Dc tha'y trung d i e m P cua M N nhm tren ( T I ) I goc • • "•• A B keo dai ve phia A Nhi/ vay a'// a v i b' // b Ta c6 a', b' ciing di qua O cd ban -'^x;' Tifdng tiT ( M A B ) ke M R M B G o i a' va b ' Ian lu^dt la hinh chieu cua a, Bai ( M C D ) n ( M A D ) = M D , nen M S M D , thi M S ( M C D ) , c! day S Nhir the a // TI b // TI va If C O neu ( M A D ) ke MS M D , thi va A B 1(71)) b tren (;t) C D ± A D (gt), C D SA (MCD) (MAD) :, , G o i ( T I ) la mat phang trung triTc cua AB d(a, (n)) = d(b (Ti)) = ^ C O , SA (P)) => C D ( M A D ) (VI A B (tiJc la (7i) qua trung d i e m O cua A B Khatuj T h i du 3: Trong (P) cho hinh vuong A B C D Qua A vc A x _L (P) Lay M thuoc T h i du 2: Trong khong gian cho hai diTcJng thang cheo a va b M va N I>, f MTV UVVII i A x ' , cat A D va A B keo dai tUdng ufng tai S' va R' K h i de dang thay rSng r la trung diem cua S'R' (do A x ' la phan giac cija goc doi dinh vdi goc BAD) Trong mat phang A x , diTng diTdng tron diTcfng k i n h S'D cat A x (chang han cat ni^a dirdng thang A x phia tren) tai M ' = > S ' M ' I M ' D ' ' 267 Boi diCCincj MSG IRnh hoc khdng gian - Phan Huy Khni Do A B J_ (Ax) r:i> D C _L (Ax) (vi D C // A B ) Cty TNIin DC i S'M' = > M ' S ' ± ( M ' D C ) va M ' S ' n ( P ) = S' vdi H O tai diem N * noi tren Theo he thiJc liTctng Irong tarn giac vuong M ' S ' D ta c6: M'A^ = A S ' A D a , H ;f;Viv(W!;.' Viet V a y quy tich cua N la dU'dng Ih^ng A (trong (P) DUclng thang niiy vuong goc •••"^ ' V i A R ' = A S ' A B = A D =:> A S ' A D = A R ' A B MTV D\ Kluing ' c i (1) T h i du 5: Trong mSt phing (P) cho gdc vuong xSy = 90" dinh S co djnh Doan A S = a vuong vdi (P) Tren Sx, Sy Ian liTdt lay B, C cho S B + S C = a Goi T h e o ( l ) = i > M ' A ^ = A R ' A B => R ' M ' B = 90" = > M ' Q ' M ' B I la tam hinh cau ngoai tiep td" dien S A B C Tim quy lich I B, C di dong L a p luan nhuTtren suy M ' R ' ( M ' B C ) va M ' R ' n (?) = R' theo quy luat tren ' B h' Vay quy tich I la nu'a dU'cfng thang Ax' la phan giac cua goc doi d i n h vdi B A D T h i du 4: Trong mat phang (P) cho diem O co djnh va d la dU'dng thang quay quanh O L a y S co dinh ngoai (P) co hinh chic'u tren (P) la H, vdi H^Q Qua S di/ng du'dng vuong goc vdi mat phang xac dinh bdi S va d Du'dng thang gnoir^/ "iX^ih mi.i & = B sin , Giai Trong (P) difng hmh chff nhat S B C D , , Goi I la trung diem cua A D De thay B D A B , D C A C (dinh ly ba du"dng vuong goc) ca't (P) tai N T i m quy tich cua N Giai K e S M d => H M d (dinh ly ba TM l A = IB = I C = I D = IS (VI cung I du'dng vuong goc) 'Trong ( S M H ) ke dUcJng thang vuong V a y bai toan tim quy tich I, difa vao phep vi tif V goc vdi S M Drfdng thang cat M H keo dai cf N T a co S N d (do d quy tich phang (tim quy tich D ) nhxi sau: (SMH)), V I S N S M =^ S N (S; d) Cho goc vuong xSy = 90" dinh S co djnh Trong tam giac vuong S M N , ta co; H N H M = SH^ = h^ = const Tren hai canh Sx, Sy Ian luTdt lay B, C di fc''^ B a i toan trd bai toan hinh hoc phang sau: Xet dufdng tron du'dng kinh O H co dinh (P) M la diem chuyen dong tren du^dng tron nay, N la diem nSm tren M H keo dai cho: H N H M = h^ = const T i m quy tich cua N nhat S B D C T i m quy tich cua dinh D Lay B B Bi tren Sx cho SB) = a => Bi co dinh N o i B , D v a g i a s u r B , D n S y = C, ^' ^ ^» B B D la tam giac Do D G doan B i C i H N * H O = h^ =^ N * la diem co dinh Dao lai lay D ' y tren doan B i C i , ta phai chuTng minh ton tai B ' e Sx, C Goi A la dirdng thang qua N * va A O H Sy cho S B ' + S C = a va S B ' D ' C la hinh chff nhat V i H N H M = h^ => H N H M = H N * H O e ^ That vay co the thay B ' , C'tiTdng ;"• ufng la hinh chieu cua D ' tren Sx, Sy =i> Do N M O = N N * b ma N M O = 90" X/ ( K ) C V a y quy tich cua doan B i C ] => N N * O = " ^ N e A Do I la anh cua D ph6p vi tif tam Da l a i l a y N ' ^ tren A Gia su* N ' H giao diTcIng tren tai M ' A ti so - , nen tiJf tren suy ra: => H N ' H M ' = H N * H O = h ' quy ve bai toan vuong can => B i S C , cung la tam giac vuong can => S C j = a => C i co dinh Goi N * la diem tren O H keo dai cho Do N I V T O = I S T N ^ = 90" X Do S B + S C = a, nen S B = B B , = S B , = a => B B , = S C ^' N N * M O la tiJ giac noi tiep dong cho S B + S C = a DiTng hinh chuT 2) tiJ g i a c N ' N * M ' la tiJ g i a c n o i tiep , ,„ifj B A;1 2j - > B C , A;il ->c 269 Ctij TNHHMTVDV\ Kluuig Vi(?t Boi ditfing HSG ITinh hoc khong gian - Phnn Iluy Khdi il day B2, C2 Ian lifdt la trung diem cua ABi; AC| Do vay quy tich I la doan B2C2, tuTc la du'dng trung binh cua tam giac A B i C i (B2C2 // B|C|), d day B,, Ci xac dinh nhiTtren tiJc la: B, e Sx, C, G Sy va SB, = SC| = a Thi du 6: Trong khong gian cho hai nufa du'dng lhang Ax va By vuong goc vdi va chco nhail nhan AB = a lam duTcIng vuong goc chung M, N tu"dng ifng chay Iron Ax, By cho ta luon CO MN = AM + BN 1) Gpi O la trung diem ciia AB va H la hinh chieu cua I len MN ChiJng minh MH = u;NH = v , ,;,f_^,, ':- 2) Tim quy tich cua H M, N di dong theo quy luat tren Giai , :>';• fM Ap dung dinh ly ham so cosin cac AM'H'O va N'H'O, co ^^X TiTdngtir, CO cosOH'N' Ke HE ± AN (E e AN) => HE // Ax vay HE (P) Theo dinh ly Talet, co: HE NH HE = uv (4) AM NM u+V Trong tam giac MBN, ke HF // NB, ma NB (Q) =^ H F (Q) UV Lai theo d i n h ly Talet, c o HF MH •HF = (5) BN MN u+v Tir (4), (5) suy HE = HF => H each deu hai mat phing (P) va (Q) => H n^m tren mat phing (n) la mat phang phan giac ciia goc nhi di^n tao b5i (P) va (Q) co dinh (do (n) co dinh) Ta CO hai tam giac vuong MAO va MHO b^ng nen OH = — =^ H'E' = H'F' Ap dung dinh ly Talet nhi/tren ta co the tinh diTcJc: H'E' = - ^ ^ va H'F' = -4^=^xy' = x'y (*) X +y X +y V a u' (**) : \ -v = - f uv ^ + (v'+v)(v'-v) = 0=>v' = v=>u' = u=>M'N' = AM'+BN' I) Phdn thuqn ua { Tit (*) suy u' = — , roi thay vao ("=*) co ''"v! - *t'i,v-''iji\ o f ) u^ + ^ — MH^ = + — - NH^ hay M H4^ - N H ^ = u ^ - v ^4 (1) Ta CO theo gia thiet MH + NH = u + v (2) T i r ( l ) v a ( ) s u y r a M H - N H = u - V (3) Bay gicJ tH he (2), (3) co ngay: MH = u, NH = V => dpcm ^,2_^2 in lii'Ut Do M'H'O + OH'N' = 180"=> 1) Ke OH MN Ta co OM^ - MH^ = ON^ - NH^ (vi cdng = O t f ) u'^+^-MO^ cosM'H'0 = ^ ua » V Cungtirdoco AM'AO = AM'H'0(c.c.c):^ M'H'O - M'AO - 90" , , => OH' MN' Tom l a i q u y tich c u a H l a nvta diTcJng tron tam O b a n kinh - ve (n) Thi du 7: Trong mat p h i n g (P) c h o hinh vuong ABCD canh a Tren canh AD l a y dilm M va dat AM = X (0 < X < a) TuT A difng diTdng thang Ax vuong g o c vdi (P) va tren Ax lay diem S Goi I la trung diem cua SC va H la hinh chieu cua I len CM Tim quy tich H M chay tren AD va chay tren Ax Giai s Goi O la t a m hinh vuong ABCD => 10 // SA 10 (P) Do IH CM => OH CM (dinh ly ba duTcfng vuong g o c ) Vay H n^m tren dirSng tr6n diTdng kinh OC ve (P) Bay g i d ta t\ g i d i h a n c u a q u y tich Khi M = A => H H O 11 271 Cty TNHHMTVDVVH Bdi dudiig HSG ITinh hoc khdng gian - Phan Iluij Khni Khi M = D, goi E la trung d i e m cua DC Ta phai chiJng minh ton tai B|, Ci tren d' cho (B|, d) i (C,; d) v^ g, \ A Do IE // SD ma SD DC (dinh ly ba difdng vuong goc) => IE DC tiJfC H H E ' t dirdng kinh OC) ^' , „ *• ta phai chu-ng minh ton tai M ' e AD, S' e Ax cho I ' H ' M ' C (d day V la trung diem cua S'C) That vay H ' e OE , nen CH' cttt canh AD va gia su" C H ' n AD = M ' Lay S' la diem luy y tren Ax (mien la S' ^ A) Ta c6 OH' H'C Do I'O // S'A => r O (ABCD) C M ' (djnh ly ba diTdng vuong goc) Vay quy tich cua H la mot phan tu" dirdng tron du'dng kinh OC, tiJc la cung O E , vdi E la trung diem cua CD Thi du 8: Cho hai diTdng thing d, d'cheo va vuong goc vdi Gia suf A la diem co dinh tren d Vdi moi diem B thay ddi tren d', chon C tren d' cho hai mSt phang (d, B) va (d, C) vuong goc vdi Goi B', C la chan Ccic That vay (R), gia suT: s ;ôrii!:j) 'ãAh Taco H l a triTc tam AAB,Ci (do BjB', Dao lai lay diem H ' tiTdng ij-ng Iren OE ; ( g 'j:>a Trong (A, d') gia suf A B ' , n d' = C, ,:5'-? OE (tiJc la mol phan tif di/dng Iron hinh chieu cua B, tren AC, B ' H n d' = B i ^ f , j V ,^„f,.i i., i/'f Vay gidi han cua quy lich chi la cung tron Khan dUdng cao AABC ke tiif B v^ C Tim quy tich B' va C B va C di AC], goc noi tie'p chSn nufa di/dng tron va A K l d ' ) C5n lai ta phai ch^ng minh (B,, d) (C,, d) That vay ta co IB, A I (do A I {%)) m^ IB, e (TI) \ Ta c6 AC, I I H ( v H H (A, d'), AC, B,B'I => AC, K I H B , ) - 1/ifW AC, I I B , =>IB| l ( I A C | ) = > ( I B , A ) l ( I A C , ) t i i ' a l a (B,, d ) l (C,, d) Vay quy tich B' la diTdng tron diTdng kinh A H (truT A, H) ve (R) (vdi C hoan loan tufdng tif) AI Tmyng hap 2.- A s I V i (B, d) (C d) va BIC (= BAC) chinh la goc phang cua nhi dien tao bdi hai mat phang noi tren nen BIC (= BAC) = 90" dong theo quy luat tren Giai Liic triTc tam H cua AABC chinh la A HsAsI=>B'=C'^I Goi IK la dudng vuong goc chung cua d va d' (vdi I G d,K e d') Xel hai tri/dng hdp sau: Vay triTdng hdp quy tich thu ve mpt diem I (s A) Tom lai quy tich TrUdnn hap 1: cuaB'(C')la: A.f\ Goi R la mat phang xac dinh bdi A va d', ttfc la R = (A, d') => (R) la mat phing CO dinh Goi H la triTc tam AABC, thco bai loan cd ban ta c6 IH (R), nhU" vay triTc tam H ciia A ABC la hlnh chieu cua diem I co djnh len (R) CO djnh H la diem CO djnh B, C di dong 1} Phan thuan: 1) Durdng tron duTdng kinh A H (trir A, H) ve (R) = (A, d') neu A ^ I , ) Diem nhaft A, neu A H I chieu cua C xuong M N Tim quy tich cua H Ta co (ABCD) // ( A ' B ' C ' D ' ) , ma (A'MC) n (ABCD) = M C Dao l a i , gia sur B', la diem bat ky tren dirdng tron n a y (B'l ^ A ; B ' ^ H ) B Giai vuong nen B', C nam tren diTdng tron 2} Phan duo: ' 'V >• dpng tren canh AB Gia suT C D ' c^t mat phing A ' M C tai N Goi H la hinh luon luon nhin A H co djnh difdi goc tron ve (R) v 11 dy 9: Cho hinh lap phiTdng ABCD.A'B'C'D' canh b^ng a, M l£k d i l m di Khi irong (R) co djnh, B' va C dirdng kinh A H (truf A v a H) DiTdng > M I \ , \ ' \ I \ \ (A'MC) n ( A ' B ' C ' D ' ) = A ' N d6 A ' N // MC (N e D ' C ) / D l t h a y B M = D'N \\ Goi I m trung diem ciia E C (tiJc I la tamciiamatbenB'BCC') ^ \ \ \ c 573 Boi ditdiig IISG Ilhih hoc kh6ng gian - Phan Hug Khdi Cty TNHH TacoCIJ-BC Khang ViH Giai M a t khac C I A B (do A B B C C ' B ' ) => CI ( A B C ' D ' ) Do M N ^ I ) G o i O la trung d i e m cua A C thi ( A B C ' D ' ) ma C H M N => I H M N (dinh ly ba diTdng vuong goc) M A = M C => M O A C Ta CO B M // D ' N va B M = D ' N Tirdng tir CO N O ± A C B M D ' N Ih hinh bmh hanh =^ M N n B D ' = o, d day O la trung d i e m ciaa B D ' , tiJc la O la tSm cua hinh lap phu'dng Vay Nhxi the' ( M A C ) va ( N A C ) ( A B C ' D ' ) H luon nhin co dinh mot goc vuong, nen H nam i tren diTdng tron dtfdng kinh ve ( A B C ' D ' ) G i d i han quy tich: , MTV DWH ,,y/':^.\;i;{;;y[^ (MAC) J (NAC) o Vay gidi han quy tich la diTdng tron M O N = 90" '; o N O ^ + MO^ = M N ^ ,^ 1) K h i M = A = > N = C ' = > M N = A C = > H = H| o u ^ + ^ + v ^ + ^ - ( v - u ) ^ + a ^ 2uv = a^ H s H2 2) K h i M = B => N = D ' =^ M N = B D ' N O M la goc gii?a hai m a t phang ) D o B x va D y (P) => ( D B M N ) ( A B C D ) H1H2 V i ( A B C D ) n ( D B M N ) = D B , ma A C D B A C l (DBMN) Trong ( D B M N ) , k e O K M N (tiJc la O K la chieu cao cua tam giac vuong NOM)=> A C I O K Nhi/vay O K la dudng vuong goc chung cua A C va M N , dieu c6 nghTa l a : HsO=>Hcodinh L a i CO tam giac vuong N O M , thi — ^ = — + OK^ ON^ / V Dao l a i lay H bat k i G H , H , ta phai chiJ-ng m i n h ton tai M e A B cho neu g o i N = ( A ' M C ) n ( D ' C ) thi H la hinh chieu cua C la M N That viiy, g o i M = O H n A B Ta c6 I H M N (goc n o i tiep ch^n difdng tron) => C H ± M N (dinh ly ba du^dng vuong goc) Vay OM^+OM^ , V 2^ + a — v^ + u V a +— + a^ — i - = O^^+OM^ OM^ OM^OM^ J v V + ^ ( v + u2) + ^ (2) nia ••ijfy Theo (1) thi u^v^ = — thay vao (2) c6: OK^ - - v^ + u ^ + a ^ quy tich cua H la nufa du'dng Iron difdng kinh H,H2 ve => O K = (ABC'D') T h i d y : Trong (P) cho hinh vuong A B C D canh a H a i nuTa dufdng t h i n g Bx, D y vuong goc v d i (P) va d ciing mot phia d o i v d i (P) M , N la hai diem tli dong tren Bx, Dy Dat B M = u, D N = v = const => H K = const Phdn thuan: Tiir chiJng m i n h tren suy m a t ph C H ± BP phai chtfng minh rang ton tai M ' e Bx, N ' Dy sac cho dong thcJi thoa man Lai C O BP C'E BP J (C'H'E) dieukiensau: =>BP1H'E '\ ' 1) (M'AC) (N'AC) • ' • '' ' ' ' ' ' - OAH),,^,/.;0 • '^nVA /•"^AH^ ^ / ' I • 2) OK' la dirdng vuong g6c chung cua AC va M ' N ' (O e AC, K' e M ' N ' ) Trong mat phang (n) = (Bx, Dy) ve tiep tuye'n vdti nufa dU'dng tron dU'dng kinh BD tai K' Tiep tuyen c^t Bx, Dy liTdng iJng tai M ' , N ' Do (71) (P) nen D B => A C l (TI) (Ji) '' ' ôã*' '' > Trong ( A B B ' A ' ) , ta c6 E nhin B H ' co' djnh du'di mot goc vuong, nen E nam tren diTdng tron difdng kinh B H ' ve mat phang ay Bay gid ta tim gidi hiin ciia quy tich Khi P = A, thi E = H , ; P = A ' , thi E = H , (H, la hinh chieu ciia H ' tren AB, H2 la hinh chic'u ciia H'tren A'B) n (P) = DB ma AC AC OK' Vi the E chi chay tren cung tron H j H j Ta phai chii'ng minh ton tai P' G AA' Hicn nhien Iheo each duTng thi: cho E' la hinh chieu ciia C la BP' OK' M ' N ' , vay OK' chinh la diTdng That vay (BAA'B') gia sij-BE' n vuong gdc chung cua AC va M ' N ' AA' = P' Do H ^ E ^ = 90" (goc npi Uep (O e AC, K' chan nu'a diTdng tron), nen H'E' E'B tai G M'N') De t h a y N ' O G AC; M ' O G AC hTOM la goc ph^ng tao bcli hai m a t phing (M'AC) va (N'AC) Trong mat phang (n), theo tinh chat cua hai tiep tuyen xuatphattir mot diem M ' B va M ' K t a CO K m i ' = N r O B Lap luan lifdng ty c6: N ' O K ' = N'OD Tur theo tinh chat cac diTcfng phan giac cua hai goc ke bu suy ra: N ' O M ' = 90" =^ ( M ' AC) G (N • AC) Tom lai quy tich chan diTdng vuong goc chung K la nfe diTcIng trbn tarn O ban kInh R - — (iijrc la niira di/dng tron di/cJng kinh DB) diftig (n) = (Bx, By) "iifiH fi\j • J * ' ' l i j ; < ( U ,,J BP' C E ' n Vay quy tich cua E la cung tron H , H cua dU'dng tron dU'dng kinh B H ' vc mat phing (BAA'B') * •' ' Thi du 12: Cho hai nuTa dU'dng thang chco Ax va By M va N la hai diem thay dd'i tren Ax va By ''''' ^ < '^ 1) DiTng mat phang (P) qua By va song song vdi Ax DU'dng thang qua M va song song v d i A B cat (P') tai M ' , „ , 2) Goi I la trung diem ciia M N , J la trung diem cija M ' N , tim quy tich J va I hai tru'dng hdp sau day: a) AM = BN ' b) A M + BN = 21 vdi / la so diTdng cho trirdc ^^R&.•^ • ' Giai ) Vdi hai diTdng thang A|, A2 song song hoac cat nhau, ta se diing ky hieu (A,, A2) chi mat phang xac djnh bdi A; va A2 Qua B ke Bx' // Ax ^ Bx' co dinh va (P) = (By, Bx') ' ' Boi dKcJiig IISG Hhili hoc khong By ^ x'By X' J B N > B N , 2) X e t trtfoTng hgtp A M = B N Ta CO B M ' = A M Vm T r e n B x ' lay M , , B y lay N , cho B M , = B N , = / => M , , N i co dinh Nhir vay quy tich cua M ' la di/tfng thang Bx' // A x 'Hg hhang Ta CO M , B N i la lam giac can Dao l a i lay M " Iren B x ' T i f M ' mat phang ( A x , B x ' ) ke diTcJng // • Tim quy tich J M i V DVVil va chii y la B A co dinh, do J chay tren doan M J ' N J tifdng « u'ng // M , N | MJ e Ot co dinh // Bd V a y I nam V i vay neu lay O A = OB , tV M la du'dng vuong gdc chung cua A va d, V a y quy tich M la mSt tru trdn xoay Qua J ' ke day cung E ' F ' A B => J ' la trung d i e m cua E ' F ' or ; ;) Qua A difng mat phang Q vuong gdc v d i A, nhU' vay Q cat mat tru theo difdng trdn ( T t ) T i f B va C ke B B ' , C C cung vuong gdc Q Ke 11' Q nhif vay cd IC = I ' B ' => , r A C ( d day O, l a c t a m ciia du'dng trdn n) V a y I ' nam tren phan du'dng trdn du'dng kinh , A nam du'dng trdn n (ma ta se goi no • O'lisr (?) M , N | // E M ' , ma E M ' , ( M , M ' , ) Trong (SAB) gia sijf S r n A B = J' •SrO'=90": MM' Trong mat ph^ng xac djnh b d i A va M M ' , tif M N i // E M ' , => M i N , A V i cua la tam difdng trdn ngoai liep cua ASE'F' V i r nam tren I j l j gdc chung cua A va d ( M e d) T i f M ke T h a t vay M M ' , J (?) => M ' , thuoc difdng trdn chuan N o i E M ' , phai difdng trdn day cho E ' F ' A B va I ' ( S E ' F ' ) n (SO') iVr; chung tren di ciia A va d| Ta C O SJ' E ' F ' ( v i J ' E ' = J ' F ' ) ma OJ' E ' F ' " tai mot du'dng thdng di qua O va M | cho M | la chan difdng vuong goc L a y r luy y t r c n cung trdn 1,12 ciia difdng i r o n dufdng kinh SO (vc " trU'dng h d p E = N vay quy goc -^^ ^ ''"A m '(0 1X/ ill (SE'F') 283 Boi duaHuj llSd IIiiili hoc khong gian - Phnn IIiu/ Khni Ciy rmni Khan,) Vict => I nam Ircn phan mat try tron xoay Ta phai chtfng minh ton tai B ' G Ox, C nhan difdng 'f lam chuan giac A B ' C Trong ( I x ' y ' ) ihiTc hien phep diTng cd ban: Dyng doan thang B|C| Dao l a i lay Ii bat ky nam tren philn mat (Bi e O x ' , C | G O y ' ) s a o c h o G ' la trung diem cua B|Ci tru ay K e I i l ' i vuong goc ( Q ) Gia su" Trong ( O x y ) gia sur A B | n Ox = B ' A l l cat (n) tai B ' , va C ' , TuT B, va C, Trong (Oyz) gia suT A C , n Oy = C difng hai du'dng vuong goc vdi ( Q ) cat V i ( I x ' y ' ) // ( x O y ) => B|C, // B ' C A l l t a i B ' l va C ' l R6 rang B|C| nhan Ii Gia si^ A G ' n B ' C quy tich I la phan mat tru tron xoay => M ' la trung d i e m cua B ' C nhan '(^ l a m difdng chuan d i e m A , B, C G o i G va H tu'dng tfug la tam va triTc tam cua tam giac A B C T i m qy tich G va H ne'u A co dinh tren Oz, va B, C chay tren Ox, Oy i; 1) T i m quy tich G '.^.B^ ^^ „„„•,,,„,,• AG' Da thay A M ' tam giac A B ' C T h i d u 18: T r e n ciic canh Oz, Ox, Oy cua tam dien vuong Oxyz, Ian lu'dt lay ba Giai ») G o i H la trufc tam tam giac A B C CO O H (ABC) ^ O H J H A Vay H n k m tren mSt cau du'dng kinh O A giac A B C tam d i e n vuong Oxyz, nen H nam tren // Oz va theo dinh ly phan mat cau difdng kinh O A g i d i han (Oxy) => GE bc(i hai m a t Oxz va Oyz, ttfc la H n^m T a l e l , la c6 GE = ^ O A tren mot phan li/ mat cau ay Dao, l a i lay H ' y Ihuoc mot phan W Nhir vay G each ( x O y ) co dinh mot khoang m a t cau ay Ta phai chti^ng minh ton ttii = ^ O A = const, vay G nam B e Ox, C e Oy cho H ' la trifc tam tren mat phang (TI) // (xOy) va di qua AABC d i e m I tren O A (trong = ^ O A ) Gia sijf A H ' n ( O x y ) = E Trong (xOy) ^ - ' ' qua E ke diTdng thang vuong goc Gu'/i hqn quy tich: M l khac G ben Theo dinh l i ba diTdng vuong goc ta co phai nam tren phan m i l l phang g i d i han •A) A bSi he mat Oxy, Oyz ciJa goc tam dien ' Dao lai gia suT G ' la d i e m y "phan tufmatphc4ng"7tlx'y' A E B'C B ' C ( O A E ) => ( A B ' C ) ( O A E ) Do H ' n ^ m I r c n mat cau difdng kinh O A => O H ' A H ' => O H ' A E =:> O H ' ( A B ' C ) (VI ( A B ' C ) n ( O A E ) = A E ) => H ' la triTc tam B, phan tu"' mat phang (n) I x ' y ' ( l y ' // Oy; R: y' vuong Ta co the hinh dung la " m o t OE Dufdng c^t Ox t a i B ' va Oy tai C goc tam dien vuong Oxyz vay G Ix'//Ox) = - => G ' 1^ tam AO Do G'u'n han quy tich: V i H n^m goc GE AI xac dinh nhi/ tren thi G n a m tren trung tuyen A M cua lam Kc ''i; Nhif the quy tich tam G cua A A B C la " m o t phan tir mat p h a n g " I x ' y ' ta G o i G la l a m lam giac A B C vi A G = 2GM ; Oxyz la goc tam d i e n vuong dinh O nen y^ny : e Oy sac cho G ' la l a m da = M => M ' B ' = M ' C (do G ' B , = G ' C , ) l a m trung d i e m va B|, C| G ( S ) V ; ) y " MTV DVVIl AAB'C V a y quy tich tri/c t a m H la m o t p h a n l\i m a t c^u diTdng k i n h O A xdc dinh O M' C nhirtren! •7 linis i i i t i JiBi)*: B 285 Chrfrfng DifcTng thang va mat phang khfing gian quan h§ song song I Tom tat li Ihuyg't II Cac dang toan Loai Cac bai toan dai ciTdng ve diTcfng thang va mat phang Loai Ciic bai toan ve thiet dien 15 Loai Cdc bai todn ve tinh song song cua difclng thang va mat ph^ng 25 ChrfdngZ Quan hv'vuong goc ' I Tom t^t ly thuyet 32 ' II C^c bai toan ve khoang each 34 A Khoang each tif mot diem tdi mot di/dng thang, hoac tiif mot diem tcfi matph^ng : 34 B Khoang each giffa hai diTitng thang cheo 42 III Cac bai toan ve goc khong gian 53 A Bai todn ve goc giffa hai di/dng thang cheo 53 B Bki toan ve goc giffa difdng thang va mat ph^ng va goc giffa hai mat phing 56 IV Suf dung phifdng phap tpa giai cac bai loin ve khoang each \;i goc khong gian 63 V The tich cua khoi da dien 80 A Tinh the tich blng each sff dung trffc tiep cac cong thffc ve the tich 81 B Tinh the tich bang each sffdung the tich ciia cac khoi da dien khdc 93 C Bai toan so sanh the tich 105 D Cac bai toan lien quan den the tich 119 E Sur dung phffdng phap the tich de tim khoang each 136 VI Cac bai toan ve quan he vuong goc 144 A Cac b^i toan ehon loc ve quan he vuong g6e 144 B Cac bai toan chu'ng minh tinh vuong goc cae de thi tuyen sinh mon toan 157 C Ciic bai toAn ve thiet dien lien quan den tinh vuong goc 162 Chrfcfng KhO'i iron xoay 170 §1 Hlnh cau 170 A Tom tftt ly thuyet 172 B cac bai loan chpn loc ve hinh cau C Nhin lai cac bai toan \6 hmh cau cac de thi tuyen sinh vao dai hoc cao ding § Hinh tru.hmh n6n 220 220 A T6m t^t ly thuyet 222 B Cdc dang toan ccf ban C Cde bai toan phoi hcfp giffa hinh tru, hinh non vdi hinh cau va 228 cac khoi da dien 235 Chtfofng MOt s6' chuyfin de dac biC't g j 235 Chuyen de Hinh tff dien 243 251 §1 Tffdi^n vuong -^jfr-264 §2 Tff dien trffe tam §3 Tff dien gandcu Chuyen de Cac bai toan qui tich hinh hoc khong gian ¥1 ... .A:Aabc S, + S2 + S3 + S d, a, 33 DSu bang xay - — - — b b , b , : "2 "7, (2) 2( S, + S2 + S3 + S) abc(S| +S2 + S - S ) _ abc(S| + Sj + S3 - S) (S, +S2 + S ) ' - S M 2( 2S,S2 +2S,S3 +2S2S3) : i... sin CO Vay -^^ sin a 3V - siny o sin a O2R^ +20 M2+40D.0M>02( 0D^+0M2 +2. 0I5.0M)>0 ,2 2. (0D + M ) > i.fi _a 2SjS^ 3a Vm Tir ( ) , (3) suy => dpcm 2S2S|Sina Khanfj V i O = H, nen O A + O B... - 2ab + a^ + SA^ = S r f + A H ^ • b ' - 2SH.r = b^ - 2ab + a^ (do S r f + A H ^ = SA^ = b^) D o SH = V S A ^ - A H ^ = Thay (2) v a o ( l ) v a cd hu2 i I r = J 3b r= a(2b-a) 2. SH (1) (2) a(2b-a)73