Tran Viet Cudng Tap chi KHOA HOC & CONG NGHE !20(06) 207 211 KHAI THAC MOI LIEN HE G I C A HINH HOC XA ANH VOI HINH HOC SO CAP TRONG DAY HOC N 6 I DUNG HINH HOC 6 TRlTCfNG PHO THONG Tran Viet Cirfrng[.]
Tran Viet Cudng Tap chi KHOA HOC & CONG NGHE !20(06): 207-211 KHAI THAC MOI LIEN HE G I C A HINH HOC XA ANH VOI HINH HOC SO CAP TRONG DAY HOC N I DUNG HINH HOC TRlTCfNG PHO THONG Tran Viet Cirfrng Trudng Dai hpc SIT pham - BH Thai Nguyen TOM TAT Bai bao nay, chiing toi de cap tdi viec khai thac moi hen he gi&a hinh hgc xa anh vdi hinh hpc so cap, dung cac kien thiic ciia hinh hgc xa anh nham soi sang, dinh hudng cho Idi giai so cap ciia bai toan hinh hgc da cho hoac khai thac moi lien he giiia chiing de sang tao cac bai toan hinh hpc mdi chucmg trinh thong Tir khoa: Hinh hgc xg dnh hinh hoc sa cdp, dgy hgc, gido vien, hgc sinh DAT VAN DE Chiing ta da biel, tir mdt khdng gian Afin la eo the xay dung duge mdt md hinh ciia khdng gian xa anb bang each them vao khdng gian afin nhiing "diem vd tan" Ngugc lai, neu ta CO mot khdng gian xa anh thi bang each bd di mot sieu phang nao dd (xem nhu mgl sieu phang vd tan) ta cd the xay dung phin cdn lai mgl md hinh xa inh ciia khdng gian afin hoac mo hinh xa inh ciia khdng gian Euclid Nhu viy, giua khdng gian afin, khong gian Euclid va khdng gian xa anh cd mdi quan he mat thiel vdi Do dd, giua hinh hge afin (HHAF), hinh hgc Euclid va hinh hgc xa anh (HHXA) Cling ed su lien quan vdi Khong gian Euclid hai chieu (E^) va khdng gian Euclid ba cbieu (E^) duge trinh bay d trudng Tmng bgc phd Ihdng (T-HPT) la nhii-ng khong gian afin theo thii ly lien ket vdi cac khong gian vecto Euclid bai chieu E~ va ba chieu E\ Bai hao nay, chiing tdi tap trung vao viec nghien ciiu mdi lien he giiia HHXA vdi HHAF va hinh hoc Euclid nham nghien ciiu, khai thac va van dung mdi Hen he giiia ndi dung HHXA vdi ndi dung HHSC day hgc hinh hgc a trudng phd thdng Qua dd, giup eho ngudi giao vien (GV) loan d Irudng thong va sinh vien su pham loan hieu ro dupe ban chit, cdi ngudn cua cac kien thiie ciia HHSC d trudng phd thdng, cung nhu thiy Tel 0978 626727 Email i dupe mdi quan be giira ndi dung kien thuc hinh hgc cao cap duge hgc d cac trudng su pham vdi ndi dung kien tbiic HHSC d trudng phd thdng NOI DUNG NGHIEN CUtf Tu- ket qua ciia HHAF suy ket qua ciia HHXA Gia su ta cd mdt dinh ly ve cac ddi tugng nao dd cua khdng gian afin Bang each them vao khdng gian afin dd cac diem vd tin, la duge mgl khdng gian xa anh, nhiing ddi tugng ciia khong gian afin trd ddi lugng cua khong gian xa inh va dinh ly da cho trd mgl dinh ly ciia HHXA Do ta chi cd mgt each la them cac diem vd lan vao khdng gian afin nen tir mdt dinh ly HHAF la ehi suy duoc nhat mgl dinh ly ciia HHXA Bang each ta co the suy mdt kel qua cua HHXA nhd mdt kel qua da biel ctia HHAF VI du: Ta da biet dinb ly sau ciia HHSC' "Trong mot hinh binh hdnh, cdc dudng cheo cdt tgi trung diem moi duang" Neu them cac diem vd lan vao mat phang afin Ihl cac canh song song ciia hinh binh hinh deu cd diem chung la diem vd tan Do dd, hinh binh hanh trd Ihanh hinb bdn canh loan phin cua mat phang xa anh Trung diem cua mdt doan thing se Ird Ihanh diem ciing vdi diem vd tin (yen dudng chiia doan thing dd) lien hap diSu hoa vdi hai diu miit cua doan thing da cho Do dd, djnh ly noi tren ve hinh binh hanh se trd ihanh mdt djnh ly ciia HHXA ve hinh bdn canh loan phin ma la da biet: "Trong moi iong2Q06^;gmail ci 207 Tran Viet Cudng Tap chi KHOA HOC & CONG NGHE hinh bon canh todn phdn, cdc dinh doi dien nam tren mot duang cheo vd cap giao diim ciia duang cheo vai hai duang cheo lgi lien hop diiu hod "Bang each nay, ta cd the dua viec giai mdt bai toan ciia HHSC bang viec giai mgt bai toan luong ling theo kien thiic ctia HHXA Ndi each khac, ta cd the sit dung cac kien thiic ciia HHXA de "soi sang" cac kien thiic cua HHSC Vi du: Tren mot tiip tuyin t cua mgt duang tron (O) lay hai diim A vd B doi xung vai qua liep diem T Tit A vd B ke hai cdt tuyen APQ, BBS cdt duang Iron (O) ldn luat tgi P, Q vd R, S Goi M, M', N, N' tucmg ung la cdc giao diim cua PR, QS, PS QR vai t Chimg minh rdng T Id trung diim cda cdc doan thdng MM' vd NN' Chirng minh 120(06): 207-211 khac di, chiing xac dinh mgl chum dudng cong bac hai (C) (Hinh 2) Trong chum cd mgt dudng cong khdng suy bien la dudng trdn (O) va ba dudng cong suy bien, dd la ba cap dudng thing (PQ, RS); (PR,QS) va (PS, QR) chiia ba cap canh ddi dien ciia hinh lii diam{P,Q,R, S} Theo dinh ly Dodac II, dudng trdn (O) va ba cap dudng Ihang ndi tren xac dinh tren tiep tuyen t tai T ciia dudng trdn (O) cac cap diem tuong ling (T, T), (A, B), (M, M') va (N, N') cua mgt phep bien doi xa anh ddi hgp loai hypebolic tren I Vi (^,5,7",oo) = - I = (5,^,7',oo) nen ta cd {M,M',T,'x>) = {N,NiT,'xi) = {A,BJ,a:>) = -\ Suy ra, T la trung diem cua cac doan thing MM' va NN' Cdch (Su dung kien thuc ciia HHSC) Dung cat tuyen AR'S' ddi xung vdi BRS qua OT (Hinh I) Theo tinh chat cua phep ddi xung true OT ta cd SS' // AB va AS = BS' (1) Suy ra, tii giac ABS'S ia hinh thang cin Do dd, ZM'AS = ZMBS' (2) Tu- ket qua cua HHXA suy cac ket qua ciia HHSC Hinhl Do ZS'AB = ZS'SB - ZS'PM nen MAPS' la lii giac ndi liep Do dd, ta cd ZAMS' = ^PG = ZS'SQ = ZSM'A (3) Tir (I), (2) va (3) ta cd AM'S'A = AMSB Suy MA' = BM - > M T = MT hay T la Irung diem MM' Chiing minh tuong ty, T la Irung diem NN' Cdch (Sic dung kiin thuc ciia HHXA): Bdn diem phan biel P, Q, R va S la cac dicm chung cua mdt chum dudng cong bac hai Ndi 208 Gia sii cd mdt dinh ly ve mdt ddi tugng nag dd khdng gian xa inh Bang each bo di mpi sieu phang nao dd ta duge mdt khong gian afm va dinh ly ndi tren se Ird mdt dmh ly cua HHAF Do cd IhS bd di bit ky mdt sieu phang nao dd nen lir mdt k£t qua HHXA, la cd the thu duge nhiSu Vk qui khac lihau HHAF Vi du "Niu tam gidc ABC ngogi tiip mot duang conic (S) thi cdc duomg thdng ndi dinh ciia tam gidc vai tiip diim trin canh ddi dien se dl qua mot diem " Tren hinh ve ta cd cac dudng thing AA', BB', CC ddng quy lai di6m O - Neu ta chgn dudng thing B ' C la dudng thang vd tan thi dudng conic (S) trd ihanh mgl dudng Hypebol voi hai dudng tiem can Tran Viet Cudng Tap chi KHOA HOC & CONG NGHE la AB va AC Khi dd, ta cd AB // OC va AC // OB Do dd, ABOC la hinh binh hanh vdi A' la giao diem cua hai dudng cheo Suy BA = AC Do dd ta di den ket qua sau ciia HHAF "Hai du&ng dim can cda mot duomg Hypebol chdn trin mot tiep tuyen bdt ky mot doan thdng ndo md tiip diem chinh Id trung diem " (Hinh 4) A 120(06): 207-211 Cung tir mgt bai loan ciia HHXA cd the suy nhieu bai loan eua HHAF nen bing each chgn sieu phang vd tan mdt each thich hgp ta cd the chuyen mdt bai toan cua HHXA mgt bai toan cua HHAF ma each giai de Ihuc hien han Vi du Chung minh rdng: Trong mgt hinh bon canh todn phdn tren moi duang cheo hai dinh doi dien vd hai diem cheo liin hap diiu hod vai Ta cd the giai bai toan bang cdng cu ciia HHXA- Tuy nhien, d diy chiing la sii dung md hinh afin cua khdng gian xa anh de giai bai loan - Neu la chgn dudng thang BC lam dudng ihang vd tan thi dudng conic (S) trd mgl dudng Parabol ma AA' la mgl dudng kinh, AB'OC la mdt hinh binh hanh Do do, ta cd kel qua sau: "Neu fir diim A ke hai tiep luyen AB vd AC vdi mot Parabol thi dudng kinh cita Parabol liin hap vdi phuang xdc djnh bai vecia BC si phdi di qua A " (Hinh 5) Hinh Chgn sieu phing vd tin P " ' di qua hai diem C, C va khdng di qua mdt dinh nao khac niia ciia hinh bon canh loan phin Khi dd, AB // A'B', AB' // A'B Suy ra, ABA'B' la hinh binh hanh ciia khdng gian afin A" Theo ket qua ciia HHAF ta cd diem cheo D li trung di6m ciia AA' va BB' Vi vay, diem D ciing vdi diem E vd tan lien hgp dieu hoa vdi hai di^m A va A' Tren dudng cheo BB', diem D cimg vdi diem vd tin F lien hgp dieu hoi vdi hai dilm B va B' Do dd, la cd (AA'DE) = (DAA') = -1 va (BB'DF) = (DBS') = -1 Viec nim viing kien thiic ciia HHXA, van dung mdi quan he giua HHXA vdi HHAF chiing la cd the dinh hudng eho ldi giai so cip ciia nhiing bai loan afin Vi du: Ggi H Id true tdm ciia tam gidc nhgn ABC Qua C dimg cdc tiip tuyin CP, CQ vdi dudng tron (O), du&ng kinh AB (P,^ Q Id cdc tiip diem) Chung minh rdng ba diem P, Q vd H thdng hdng Lai gidi 1: {Theo goc cua HHXA) Gpi D = BC n AH, E = CA n BH, F = DE n AB, = 209 Tran Viet Cudng 120(06): 207-211 Tap chi KHOA HOC & CONG NGHE BE r^ CF, K = AD n CF Xet tu giac toan phan ABDECF ta cd [ADHK] = [CFKI] = [BEIH] = -I Suy ra, H lien hgp dieu hoa vdi I va K ddi vdi dudng trdn (O) Do dd, IK la dudng ddi eye ciia H, nen C lien hap vdi H ddi vdi dudng trdn (O) Mat khac, PQ la dutmg ddi cue cua C, suy H thugc PQ hay P, Q va H la ba diem thang hang phang vd tan, ta cd the co nhiiu bai toan ciia HHAF khac ma cac k€t qua ta cd the suy tir nhiing kk qua da biel HHXA Ket hgp ca hai each lam ta cd the tir mgt bdi todn sa cdp suy nhieu bdi todn sa cdp khde C Hinh Ta thay, PQ la dudng ddi cue cua C, ma C lien hop vdi H ddi vdi dudng Iron (O), nen H thugc PQ, suy H, P, Q Ihang hang Viy de chung minh H, P, Q thang hang, ta chung minh H Ihudc dudng ihang PQ Dieu ggi y cho la thiy H nam Iren true dang phuang PQ cua hai dudng trdn nao va la cd the dua ldi giai so cap bai loan tren L&i gidi {Theo goc cua HHSC): Ta cd, cac diem C, P, F, O va Q cimg nam Iren dudng trdn (o) dudng kinh OC (hinh 8) Do dd, ta cd: /'(H)/(co) = HC.HF /'(H)/(0) = HA HD = HB HE Mat khac, H la true tim ciia AABC nen ta cd HA HD = HB HE = HC.HF Suy P(H)/(a)) = P(H)/(0) hay H Ihudc true dang phuang PQ cua (w) va (O) Viy P, Q va H la ba diem thing hang Sang tao cac bai toan moi Tir mgl bai loan ctia HHAF la cd the suy mgl bai loan cua HHXA bang each bo sung them vao khdng gian afm nhiing diem vd tin thugc mdi sieu phang vd tin Ngugc lai, tir mdt bai toan cua HHXA, bang each chgn cac sieu phang khac ddng vai Ird sieu 210 Viec nam viing kien Ihiic HHXA, ngudi giao vien (GV) loan THPT cd mdi minh dat "mau md" de sang tao cac bai toan cho hgc sinb ciia minh luyen tap Do dd, mgt GV THPT vdi kien thuc ve HHXA duge trang bi cdn la sinh vien d trudng Su pham cd the de dang dua mdi sd bai loan HHSC d Irudng phd Ihdng ve bai toan ciia HHXA, diing kien Ihiic HHXA soi sang, dinh hudng cho ldi giai so cap ciia bai toan da cho, hon the niia lir bai loan cua HHXA luang ling, GV dd cd the tao duoc nhieu bai toan so cip cd mdi lien he vdi bai loan ban diu theo dudng: , Afm hoii Tu bai toan E r bdi todn A AUn hoa xa anh hoa , r Bai todn P^ , Cac bai toan A' Tmc chu^ hoa ^ , , '' Cdc bdi todn E\ Dd la sy the hien cua mdi lien he chit che giiia toan hgc phd thdng vdi toan hpc cao cap theg cac cgn dudng: Toan hgc cao cap -> Toan hgc phd thdng hodc Toan hgc phd ihdng -> Toan hpc cao cap -> Toan hgc phd thdng Tit nhien, nhung ngudi cd the di theo dudng chi phii hgp la nhung sinh vien su pham - nhiing ngudi GV tuang lai va nhiing GV dang true liep giang day d cac Irudng phd Ihdng Lam dupe nhu the, sinh Tran Viet Cudng Tap chi KHOA HOC & CONG NGHE vien se nam siu sac cac kien thuc loan cao cap, thiy duge mdi lien he vdi toan hgc phd thdng, gdp phin lam tdt khiu chuan bi nghe nghiep sau va chac chin se cd ket qua Idt cic ki thi ciia minh Cdn ddi vdi nhiing GV phd thdng, di theo dudng la mdt each de ning cao trinh chuyen mdn nghiep vu cua minh, nang cao hieu qua day hge va tat nhien nhiing hgc sinh duge hgc nhiing ngudi Ihay nhu vay se cd nhieu eg hdi duge luyen tap, khic siu va duge khai thac, md rdng kien thiic tir mgl dang toan d i cho KET LUAN Tir nhiing phan tich tren, cho chiing ta thay: Giua npi dung HHXA duge hgc d cac trudng Su pham va ngi dung HHSC duac bgc chuong trinh phd Ihdng co mdi quan he mat Ihiel vdi Do dd, neu ngudi GV biet each khai thac, van dung linh boat mdi quan he vio viec day hgc hinh hgc d phd thdng thi se gop phan nang cao hieu qua day bgc cho hgc sinh Hon nUa, de ning cao ehal lugng ngudi GV tuang lai, qua trinh giing day, cac giang vien bg mdn hinh hgc can danh thdi 120(06): 207-211 gian de phan tich cho sinh vien thiy dupe mdi quan he giiia ndi dung HHXA vdi ndi dung HHSC chuang trinh phd Ihdng, qua dd giiip cho cic sinh vien su pham toan hieu rd duge ban chat, cdi ngudn cua cic kien thiic ciia HHSC d trudng phd thdng, ciing nhu thay duge mdi quan he giira npi dung kien Ihue hinh hgc cao cap dupe hpc d cac trutmg su pham vdi ndi dung kien thiic HHSC d Irudng phd thdng TAI LIEU THAM KHAO Pham Binh Do (2006), Bdi tap hinh hgc xg dnh, Nxb Dai hgc Su pham Van Nhu Cuong (1999), Hinh hgc Xg dnh Nxb Giao due Van Nhu Cuong, Ta Man (1998), HHAF vd hinh hoc Euclid, Nxb Dai hgc Qudc gia Ha Npi Tran Viet Cudng, Nguyen Danh Nam (2013), Gido trinh HHSC, Nxb Giao due Viet Nam NguySn Mong Hy (1999), Hinh hgc cao cdp, Nxb Giao due Nguyin Thj Minh Ykn (2006), Xdy dimg mdi so chuyen de "cdu noi" giiia hinh hgc cao cdp d trudng Cao ddng Su phgm vdi hinh hgc d phd thdng nhdm tdng cudng dinh hudng su pham cho sinh viin, Luan van Thac sT Khoa hpc giao due SUMMARY APPLICATION ON THE RELATIONSHIP BETWEEN PROJECTIVE GEOMETRY AND PRIMARY GEOMETRY IN THE GEOMETRY TEACHING AT THE HIGH SCHOOL Tran Viet Cuong College of Education - TNU In this paper, we refer to the application on the relationship between projecUve geometry and primary geometry, using the knowledge of projective geomeny to lighten, to guide the primary solution of given geometiy problem or application on their relationship lo create new geometry problems in school programs KeyvtorA- projective geometry, primaiy geometry, teaching, teacher, student Ngdy nhgn bdi-31/1/2014; Ngdyphdn bi^n:24/2/2014; Ngdy duyit ddng: 09/6/2014 Phdn bien khoa boc: TS Do Thi Trinh - Tnrdng Dai hgc Suphgm - DHTN Tel: 0978626727 Email lranvietcuong2006@gmail coi ... siu va duge khai thac, md rdng kien thiic tir mgl dang toan d i cho KET LUAN Tir nhiing phan tich tren, cho chiing ta thay: Giua npi dung HHXA duge hgc d cac trudng Su pham va ngi dung HHSC duac... Trong mgt hinh bon canh todn phdn tren moi duang cheo hai dinh doi dien vd hai diem cheo liin hap diiu hod vai Ta cd the giai bai toan bang cdng cu ciia HHXA- Tuy nhien, d diy chiing la sii dung. .. {N,NiT,''xi) = {A,BJ,a:>) = -\ Suy ra, T la trung diem cua cac doan thing MM'' va NN'' Cdch (Su dung kien thuc ciia HHSC) Dung cat tuyen AR''S'' ddi xung vdi BRS qua OT (Hinh I) Theo tinh chat cua phep ddi xung