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[.]
Trang 1KHAI 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
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 ra cac bai toan hinh hpc mdi trong chucmg trinh pho 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
thanh 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 nhau Do dd, giua hinh hge afin
(HHAF), hinh hgc Euclid va hinh hgc xa anh
(HHXA) Cling ed su lien quan vdi nhau
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 trong day
hgc hinh hgc a trudng phd thdng Qua dd,
giup eho ngudi giao vien (GV) loan d Irudng
pho 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 iong2Q06^;gmail ci
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 ra 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 thanh ddi lugng cua khong gian xa inh va dinh ly da cho trd thanh 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 trong HHAF la ehi suy
ra duoc duy nhat mgl dinh ly ciia HHXA Bang each nay ta co the suy ra 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 nhau 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
Trang 2hinh bon canh todn phdn, cdc dinh doi dien
nam tren mot duang cheo vd cap giao diim
ciia duang cheo do vai hai duang cheo con
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
nhau 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
Cdch 1 (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)
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 ra MA' = BM - > M T = MT hay T la
Irung diem MM'
Chiing minh tuong ty, T la Irung diem NN'
Cdch 2 (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
khac di, chiing xac dinh mgl chum dudng cong bac hai (C) (Hinh 2) Trong chum nay
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'
Tu- ket qua cua HHXA suy ra cac ket qua
ciia HHSC Gia sii cd mdt dinh ly ve mdt ddi tugng nag
dd trong 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 thanh mdt dmh ly cua HHAF Do cd IhS bd di bit ky mdt sieu phang nao dd nen lir mdt k£t qua
trong HHXA, la cd the thu duge nhiSu Vk
qui khac lihau trong 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
Trang 3la 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 ra
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
- Neu la chgn dudng thang BC lam dudng
ihang vd tan thi dudng conic (S) trd thanh
mgl dudng Parabol ma AA' la mgl dudng
kinh, con 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)
Cung do tir mgt bai loan ciia HHXA cd the suy ra 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 thanh 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 nhau
Ta cd the giai bai toan nay 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 nay
Hinh 6 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 do cua HHXA) Gpi D =
BC n AH, E = CA n BH, F = DE n AB, 1 =
Trang 4BE 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 ra H thugc PQ hay
P, Q va H la ba diem thang hang
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 ra H, P, Q Ihang hang Viy de
chung minh H, P, Q thang hang, ta chung
minh H Ihudc dudng ihang PQ Dieu do ggi y
cho la thiy H nam Iren true dang phuang PQ
cua hai dudng trdn nao do va la cd the dua ra
ldi giai so cap bai loan tren
L&i gidi 2 {Theo goc do 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 ra 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 ra
mgl bai loan cua HHXA bang each bo sung
them vao khdng gian afm nay 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 nhau ddng vai Ird sieu
phang vd tan, ta cd the co nhiiu bai toan ciia
HHAF khac ma cac k€t qua ta cd the suy ra tir
nhiing kk qua da biel trong HHXA Ket hgp
ca hai each lam nay ta cd the tir mgt bdi todn
sa cdp suy ra nhieu bdi todn sa cdp khde
C
Hinh 8 Viec nam viing kien Ihiic HHXA, ngudi giao vien (GV) loan THPT cd mdi minh dat "mau md" de sang tao ra cac bai toan cho hgc sinb ciia minh luyen tap Do dd, mgt GV THPT vdi kien thuc ve HHXA duge trang bi khi 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
ra duoc nhieu bai toan so cip cd mdi lien he vdi bai loan ban diu theo con dudng:
, Afm hoii
Tu bai toan trong E r bdi todn
2 xa anh hoa , trong A r Bai todn trong P^
AUn hoa ,
7 Cac bai toan trong A'
Tmc chu^ hoa ^ , ,
'' 7 Cdc bdi todn trong 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 con dudng nay chi phii hgp la nhung sinh vien su pham - nhiing ngudi GV trong tuang lai va nhiing GV dang true liep giang day d cac Irudng phd Ihdng Lam dupe nhu the, sinh
210
Trang 5vien 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 nay va chac chin se cd ket qua Idt
trong cic ki thi ciia minh Cdn ddi vdi nhiing
GV phd thdng, di theo con dudng do la mdt
each de ning cao trinh do 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 trong
chuong trinh phd Ihdng co mdi quan he mat
Ihiel vdi nhau Do dd, neu ngudi GV biet
each khai thac, van dung linh boat mdi quan
he do 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
trong tuang lai, trong qua trinh giing day, cac
giang vien bg mdn hinh hgc can danh thdi
gian de phan tich cho sinh vien thiy dupe mdi quan he giiia ndi dung HHXA vdi ndi dung HHSC trong 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
1 Pham Binh Do (2006), Bdi tap hinh hgc xg dnh,
Nxb Dai hgc Su pham
2 Van Nhu Cuong (1999), Hinh hgc Xg dnh Nxb
Giao due
3 Van Nhu Cuong, Ta Man (1998), HHAF vd hinh hoc Euclid, Nxb Dai hgc Qudc gia Ha Npi
4 Tran Viet Cudng, Nguyen Danh Nam (2013),
Gido trinh HHSC, Nxb Giao due Viet Nam
5 NguySn Mong Hy (1999), Hinh hgc cao cdp,
Nxb Giao due
6 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 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