立体几何证明题专题(教师版)分析

ABCDABCDBBABBBCBAB

CBABCDBBABBBB

CDABCDBCADABCD

Plm

Plm

Plm

Plm

Plm Pn

lmlm

lm

P

lmP

lm

P

lm

P

lm

/

abαβαβcc

ababab

ab

c

abc

ab

abab

PQRS

PSQRPQRS

PQSRPQRS

QR

ABCPS

ABC

P

P

QR

PS

QRPQRS

PQRS

BC

NPSNR

BCPNRS

PSQNPQNSβαβPNSαβPQRS

α

abcabcabc

ababc αβaβγbγαcCFCG AE

CF

EHFGBDEH

BDCB

CD CBDCFCG AE

CF

EHFGBDEH

BDCB

CD CBDFGBD

FG

BDEB

FB

ab

abOOaa

α

OαObb

γ

OγγαcOc

aba

γb

γ

aγ

a

ααγc

ac

ABCDEHABADFGBCCDCB

CD EFGHACAH

CGEB

FB HD

GD EH

ABADCF

CG

EHFG

EHFG

EFGHEHFG

EFGHPPEFEF

ABCPABCP

ADCPABC

ADC

ABC

ADCACPAC

EFGHACAE

CF EFAC HGACEFHG

EFHG

AH

CGHD

GD

EFGH

EH

FG

P

PABDPBCD

/

PBDEHFGBD

ABCDABCDE

ABBD

CE

ABCE

ME

BDCMCDDM

AD

ABCE

ME

BDCMCDDM

MECMECCEMECM CEME

CEME

BD

CE

ABCDABCDBD

AD

ABCDABCDAAABE

AABE

CD

BA

CDBAABE

BE

CD

AB

BE

BA

Aeq

\o\ac(△,E) ABEABE

ABCDABCDE

CDAE

BC

AB

F

EFFAEFBCBCAEF

AE

BCABCDABCDa

EF

a

AF a

a

a

AE

a

a

aaAEF

AEF

AE

BC

AEEFAF

AE

BC

AEEF aa

aαaα

lα

lα

lα

lα

lαlα aαAaα

lαlαlααlabbαaα

a

αlαlαl

αAC

BD

ABCD abα

aαb

α

ab

aαbα

ab

abb

α

aα

aba

α

bα

b

α

aαb

α

ab

aαbα

ab

/

PMABQN

abb

αaα

a

α

PMABQN

ABCD

ABEFAB

AEBD

PQ

APDQPQ

BCE

PMAB

BE

M

QNAB

BC

N

MN

ABCDABEFABAEBD

APDQPEQBPM

PE

QB

QN

BQAB

AE

BD

DC

BDAB

AB

DC

PM

QN

PEBQAPDQ

ADBK PEBQAPDQ

ADBK PQEKPMBE AP

DQPE

BQ

MB

QBPQMN

MN

BCEPQ

BCEPQBCE AQBCK

EKAEBDAPDQPE

BQDQ

AQ AP

AQBQ

QK PE

QK

PQ

BCEEK

BCEPQBCE ABEFP

PMBE

AB

M

QMPMBCE

ABEF

BCEBEAP

AMPE

MB

AEBDAPDQPEBQAM

DQ MQAD

ADBCMQBCMQ

BCE

PMMQM

PMQ

BCE

PQ

PMQPQBCE

MN

AFBCMNCDEFACDEF

ABBCBFDECF

CBF

BF

G

MGNG

MN

AFBCNGCFMGEF

MGNGGCFEFF

MNG

CDEFMNCDEF

AHDE

HADEBCFAHCDEFAH

VA－CDEFS

CDEFAH

PabP

ab

ABCDABCD

N

BD

M

BC

CMDNMN

AABB

MEBC

BB

E

NFAD

AB

F

EF

EF

AABBBDBCDNCMBMBN

MEBMNFBN

BC

BC

AD

BD

/

MEBNNFMENFDN

CN

CMDNBCMEBNNFMENFDN

CN

CMDN

MEBCADNFMEFNNMEFMN

AABBMNAABB CNBAP

BP

BP

AABBNDCNBPNB

NPMB NB

NPBMB NB

NP

CM

DN

CN

MNBPBP

AABBMNAABBMPBB

CMMB PBBMBNMB MPBB

CMMB PBBMBNMB NBCP BDBCDNCMCM DN NPDCAB

CP

DNPB

NB

MNP

AABBMNAABBPABCDABCDPDABCDPDABEFGPCPDBCPAEFGPEFG AD

H

GHFHEF

PCPDEFCDGH

BCADGHCDEFGHEFHGFH

DPDAPAFHPA

EFGFH

EFGPAEFGPDABCDCG

ABCDPDCGCGCDCDPDDGCPCD

EFPFPFPDEFCDeq

\o\ac(△,S) PEF

EFPF

GCBCVP—EFGV

abAC

BDabaαbαABαMCDαNAMBM

CNDN

ADα

EMENEbαME

ABDα

ABDMEMEeq

\o\ac(△,BD) ABD

AMMBAEED

E

AD

aαEN

ACDα

ADCENENAC

E

ADN

CDCNDNABCABCE

ACAB

CEBAEEC

BC

BC

F

F

BCAB

CEBAB

ABCCEB

ABCEF

/

ABEFE

ACAEEC

mnαllβαβ mβ

lα ml

nl mβ

nβ mβ

nl

m

αl

βαβ

mβ

lααβ

mβ

lα

mn

αll

β

l

lml

nl

l

lm

n

αβαβ

m

lαβml

nl

ABCDABCD

PQR

ABCDBCCBABBAPR

BQRQBCCB

PQR

DAC

PQR

PR eq

\o\ac(△,A)BC

PRBQ

RQ

ACACBCCBPRBCDADeq

\o\ac(△,C) ABC

PABCGGGPABPCBPAC GGG

ABC

eq

\o\ac(△,S) G1G2G3

eq

\o\ac(△,S) ABC PGPGPGABBCAC

DEF

DEEFFD

PGPDPGPEGGDE

GGABCGG

ABC

GG

ABC

GGGGG

GGG

ABC

PQ

AB

确．故填③PG PG eq

\o\ac(△,S)PG PG eq

\o\ac(△,S) GGGeq

\o\ac(△,S)

PE

PD GG

PE

DEACGGAC

GGABGGBC eq

\o\ac(△,G)GGeq

\o\ac(△,3) CABlmnαβγ

l

ml

αm

βαβαβl

αm

β

lmαβlβγmγαnlγ

mn αβlm

l

m

lγl

β

βγm

ln

mnABCDABCDMNEF

ABADBCCDAMN

EFDB

MFMF

ABCDABCDMF

AD

AD ADMF

AD

AMFDAMDFDF

EFDBAM

EFDBAMEFDB

AN

EFDB

AMAN

ANMAMANA

AMN

EFDBABCDABCDMNPCCBCCDMNPABD BDPN

DCBC

/

PNBD

BDBDPNBD

PN

ABDPNABDMNABD

PNMNN

PMN

ABD ACACABCDABCDACBD

CC

ABCDAC

ACABCDACBD

ACABAC

ABD

ACPMN

PMN

ABDαβ

AαCαBβDβEFABCD

AEEBCFFDEFβ

ABCDαβαABDCACβ

ABDCBDACBDAEEBCFFDEFBD

EFβBD

βEFβ

AB

CD

ACDβDH

DHACαβαACDHACACDH

ACDH

AHGAGGHCFFDAEEBCFFDGFHDEGBH

EGGFGEFGβEF

EFGEFβEFβABCABCDD

ACACADDCBC

ABDADDCADBCD

ABDDC

D

D

ACAD

AB

AB

OODADDCAO

DC

ADOB

AD

DCAABBO

AB eq

\o\ac(△,A)BC

OD

ABACODBCOD

ABDBC

ABDBC

ABD

BCABDBCDABD

ABC

BDCBC

ABC

ABDDO

BCDOADDC

ADADAO

ADDC OB DC AD

αβab

aαbα

ab

aαbβabαβ

aαbβabαβ

abαab

OAaOBb

OAOB

αβ

ACBCOAACOBBC

OACB

ACB

αlβαβ

abαmn

mn

ab

/

lαlα

mnαmαmαnαmαmnnαmnmn

nα

mn

mα

nα

mα

PABCDPA

ABCDABADACCDABCPAABBCE

PCCDAEPD

ABE PA

ABCDCDPA

CDACPAACA

CD

PACAE

PAC

CDAEPAABBCABCPAACE

PCAEPC

CDAE

AE

PCD

AEPD

BAPD

PD

ABE

lαβ

lαlβαβ

lαlβαβ

lααβlβαβlαlβ

lαlβαβ

αβlα

lβ

l

β

αβlαlβ

bcαβ

b

αcα

bc

b

αbc

cα

cαcβαβ

cααβcβ

cααβ

cβ

ABCABCAA

ABCACBCAAACBE

BBADECDAABB

AEECACBCACBAB

ADx

BD

xADxDE

xAE

ADEADDEAEx

D

ABCDAB

AACD

AAABACDAABBABCDABCDABCDO

BDE

AABDEC

ABAE

OEEC

AA ACACAC

BDOBDAC

AAABCDBD

ABCD

AABD

AAACA

BD

AACC

EC

AACCBDEC

AA

h

OC

OAEAE

AO

OE

eq

\o\ac(△,EA)CAEh

AC

/

EF

ECBDFDBCAB

CAN

MNBNEF

ECBDFDBCAB

CAN

MNBN

MN

EC

OCCOC

CChOC21h

OEECOEECOC

h

h

h

AA

ABCEC

ABCBDCE

CECABDM

EADEDA

BDM

ECA

DEA

ECA

EC

FDFBDCEDBBA

ECBC

EFD

DBAEFDDBADEDAMNBDNBDMECABCECBN

CABNBN

ECABN

BDMBDM

ECADMBNBN

ECADMECA

DM

DEA

DEA

ECAPAB

ABC

PAC

ABCAE

PBCEPA

ABC

EPBCABC ABC

D

DFAC

F

PACABCACDF

PAC

PA

PACDFPA

DGAB

GDGPADGDFABCPAABC

BEPC

HEPBCPCBH

AE

PBCPC

PBCPCAE

BHAEEPCABE

AB

ABEPCABPAABCPAAB

PCPAPAB

PAC

AC

PACABACABCABCABCAB

AC

BBCCABC

D

BCADCCBBCCBCM

AMMAMBC

BBCCAMMAMBC

BBCC ABACD

BCADBC

ABC

BBCCBCAD

BBCCCC

BBCCADCC

BCE

DEeq

\o\ac(△,ME) BCCDE

BCBCDECC

AA

CCDEAAM

AA

AMMADE

AMAMEDAD

ME

AD

BBCCME

BBCCME

BMC

BMC

BBCC

BA

BM

N

AABB

CNAMMANAABABeq

\o\ac(△,AC) ABC eq

\o\ac(△,A)BCABABACACACANAB eq

\o\ac(△,B)CNNCB

CNBC

BBCC

ABCBCNC

BBCCNC

BNC

CNB

BBCC

MBC

BBCC

/

BMC

BBCC

AMMAM

AA

M

MEBC

E

MBC

BBCC

BCME

BBCC

AD

BBCCADMEMEDAAM

BBCC

AMED

BBCCDEAMDE

ADME

AMEDADMEDE

AMAMCCDECCD

BCE

BC DECCAAAMAAMAMAAMMA

MBC

BBCC

ABEF

DCEF

CDFDABEADFAFABCDEBCD

AFFDADFD

AFFDADFDADFDA

BC

ADFAD

ADF

BC

ADF

CE

ADF

BCCECBCCE

BCE

BCEADF

BE

ADFFDAFDAD

eq

\o\ac(△,S)

ECB

eq

\o\ac(△,S)

ECB

VVD－EBC

DC