[PDF] Géométrie plane Points vecteurs. Le plan affine

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(A;B)ǯ(A;B)(D;C)ABCD

ǰ ȂȬȬ (AB)(CD) ǰ

AB=

DCǯ

ę ŗǯŘǯ (A;B)(B;C)

(A;C)ǻ Ǽǯ k !v (A;B)ǰk!v (A;C)C (AB) AC=k

ABǯ

!AB (A;B)ǯ

ę ŗǯřǯ(Ai) ǻi1nǼ (ki)

ǯ Aiě Ĝki Mǰ

n∑ i=1k i!MAi=!0:

O ę ǰ M

OM=1 n i=1kin i=1k i!OAi

ǰ ABě Ĝab M

!OM=1 a+b(a!OA+b!OB)ǯ (M;b+c)M (B;b)(C;c)ǯ

ŗǯśǯ Ĝf

ǰ ę !f(!AB) =!f(A)f(B)ǯ

ę f(!u+!v) =f(!u) +f(!v)ǯ

Ȃ Ȃ A !u Ȃ B

(A;B) !uǯ

řę ŗǯŝǯȂ O k ę

!OM′=k!OMM′ Ȃ Mǯ

ę ŗǯŞǯ E ǻ Ǽ

ęŗǯşǯĜEǻEǼ

Eǰ ę (M+!u) +!v=M+ (!u+!v)ǰ

(M;N) !u M+!u=Nǯ !u !MNǯ

Ȃ ǻǰ Ȃ EǼǯ

Řǯŗǻ ǼǯABC Bǯ AB2+

BC

2=AC2ǯ

ę ŘǯŘǻ Ǽǯ!u!v ǰ !u!v

!u !v!uǯ !u(!v+!w) =!u!v+!u!wǯ !AB!AC=ABAC[BAC:

AB2=!AB!ABǯ

Řǯřǻ Ȃ ǼǯAǰBǰC ǰ AC

2=AB2+BC22ABBC[ABC

(x;y) Ĝz=x+iyǯ ę ŘǯŜǯ AǰBǰCǰDȂĜaǰbǰcǰdǰ [A;B;C;D] =ca cdbd ba

ę[0;1;t;1] =tǯ

Řǯŝǯaǰbǰc Ĝ AǰBǰCǯ ȂĜz

ABC [a;b;c;z] ǻȂę Ǽǯ

ŘǯŞǯz7!az+b

cz+dǯ

ǰ z7!az+bǰ

jaj a ǯ k2 M M′ (OM) OM OM ′=k2ǰ O Ȃę ǯ Ȭ z7!k2/zǯ p Ȃ P ȂȬ p/Pǯ

Řǯşǯ O Pǰ Ȃ

Ȃ O p Ȃ Ȃ O

p/Pǯ

Ȃ Ȃ Pǯ M N

(OM)ǰ OM

ON=Pǰ Ȃ MNǯ

ŘǯŗŖǯO Ȃ ǰ MN ǰ

ȂM′N′ǯ MǰNǰM′ǰN′ OMNOM′N′

ǯ M′N′=MN

OMONǯ

OM OM ON ON ′=Pǰ OM ON =ON′ OM M ′N′ MN =OM′ ON OMON:

śDAB

CXY Z

Řǯŗŗǻ ǼǯAǰBǰCD ǯ

ACBD6ABCD+ADBC

AǰBǰCǰDǻǯǯ[A;B;C;D]>

1Ǽǯ

Ȃęǰ AǰBǰC XǰYǰZǯ XZ6XY+Y Z

XǰYǰZ Ȃę ǰ AǰBǰCǰ

D ǯ

XZ=AC

ADCDXY=AB

ADBDY Z=BC

BDCD

ȂXZ6XY+Y ZADBDCDǰ ǯ

ę ŘǯŗŘǯ E Ȃ

(;)7!ǰ ę ǻ Ǽ Ȭ ∥∥2=ǯ

Ŝ A

B C MNP G

Ĵ A Ȃ

M MABMAC ǯ

řǯŘǯG Ȃ MǰNǰP

[BC]ǰ[CA]ǰ[AB]ǰ GMBǰGMCǰGNCǰGNAǰGPAǰGPB ǯ

Ȃ [AB][AC]

Aǯ A B C HAH B H C H

ABC HAHBHCǰ H

ǯ ǰ O

ǰ O ǯ

O OAǯA

B C O

O ǻ Ȭ

řǯśǯ H

[BACǰ[CBA[ACBǰ \BAH=/2 \HBA=/2ǰ Ȃ\AHB=\BAH\HBA=ǯ S

CH (AB)ę\ASAB=[ACBǰ AǰBǰCǰSC

ȂBCǯ

A B C O MN PHAH B H C S AS B S C H A ?B C

Ȃǰ ǯA

B C H P PAP B P C P?AP ?B P ?C (CA)ǰ(AB)ǯ \PPAC\PPBC ǰ PǰCǰPAPB \PPBPA=\PCBǯ ǰ\PPBPC=\PCBǻ180Ǽǯ

PǰAǰBǰC ǰ Ȃ ǯ

ǰ PǰAǰBǰC ǯ

ǰ SAH (BC) Ȭ

ǯ P ǰ HǰBǰCǰP′A ǰ P′A

P (BC)ǯ \BP′AH=\BCH=/2ǯ

\BP′CH=/2ǰ B PP′AP′Cǰ \P′APP′C= 2\P′APP′C= 2 PǰBǰPAǰPC ǯ

Ȃǯdi=djdk

kǰ di+dj= 0ǯ

IAIBICǻ Ǽ

ABC ǰ I ǯ

řǯşǯABC ǰA′ ȂBC ǰI

IA ȂAǯ A′

şA BC II A A M K A L A

BǰCǰIǰIAǰ

(BC) M(BC)ǯ

CǯICIACǰĜA′

[IC]ȂȂǯ IA ′C A′ǯ Ȃ[CA′Iǰ Ȃ[ICA′ ȂCB

ȂCAǯ /2ǰ ǯ

ęǰ [BC]ǰM KALAA′

IIAǯ

řǯŗŖǯABC ǯ ǰ Ȃǰ

ǰ [AH]ǰ[BH]ǰ[CH]ǯ

[OH]ǰ !OH= 3!OG=!OA+!OB+!OC:

OǰGH Ȃǯ

ǰ R/2ǯ ABC

MNP ǰ

ABCO Ȃ MNPH Oǯ !OH= 3!OG Ȃ [OH]ǯ

Ȃ H 1/2

R/2[OH]ǰȂȬȬȂȬ

ǯ H ǻ Ǽ

[AH]ǰ[BH]ǰ[CH]ǯ

M ǯ ęǰ

ŗŖ A

B C O GH MN P HAH B H C S AS B S C A?B C A B CI K A H H AO A ?MΩ

Ȃ ∆ MΩǰ Ω [OH]ǰ

Ȃǯ M

bAǰM Ȃ ǰ

ǰ (BC) ǻĴ

Ȃ G 1/2 HAOOMΩǰ

OMΩ =\HAO=2(/2) =

D Aǯ ě\ADC\ADB

(MΩ) ∆ǰ (OM) (BC)

Ĝ (HA) =D Ȃ

∆ǰ D ∆∆ MΩǯ

MK2A=MDMHAǰ HA KA

ǯ A′I2=A′DA′Aǰ

(BC)ǯ Ȃ A ′D/A′B=A′B/A′AA′I=A′Bǯ A′BDA′BA ǰ

Ȃ Ȃ A′ \A′BD=\A′AB=/2ǯ

ŗŗA

B C

MNPHAH

B H C S AS B S C A?B C I I AI B I C KAK B K C L AL BL C O G ΩH

KAǰKBǰKC ǰ LAǰLBǰLC

řǯŗŘǯABC ǯ

Ȃ M Ȃ Mǯ

O Ȃ2/32/3ǯ d(M;AB) +d(M;BC) +d(M;CA) =d(M;AB) + d(P;AB) +d(N;AB) = 3d(O;AB)ǯ

řǯŗřǯABCǰMǯMA+MB>

MC M ȂAB ǯA

B C M N

MA+MB=MN+NC>MC Ȃ ǻ ANM

Ǽǯ Ȃ N [CM]ǰ ȂȬȬ

\ANC=\AMB= 2/3ǰ ȂȬȬ M ȂABǯ

ǰMA+MB+MCǯ

\AMB=\BMC=\CMAǰ

2/3ǯǰ

ABTCǰBCTAǰCATBǯ MA+MB+MC>MA+MTA>

AT

AǯĴMM

2/3 ǯ

F Ȃ ABTC

BCT

Aǯ [AFB=\BFC= 2/3ǯ [CFA 2/3ǰ

F ǯ ǰ ǰ \AFTB=\TBFC=

CT

C Fǯ

FA+FB+FC=ATA=BTB=CTCǯ

ŗřA

B C T AT B T C F

řǯŗśǯABC ǰ M ǯ

(AM)ǰ(BM)ǰ(CM)ǰ AǰBǰCǻ Ȭ (AM)ǰ(BM)ǰ(CM)Ǽ Nǰ Mǯ

Ȃ (O;H)ǻȂ

(x;y;z)7!(yz;zx;xy)ǯ

řǯŗŜǯMN ABCǰ

[MN]ǯ N

ANB Nǯ CMAMMBCNANNB Ȭ

ǯ CMACNA=CMBCNBǰ \CMAMB\CNBNA ǰ

\NAMAMB\NANBMB ǰ MAǰMBǰNAǰNB B C M N MAM B M CNAN BNC

řǯŗŝǯP ǰ

(AP)ǰ(BP)(CP) P A B C PP AP B P CQ C

ȂPȂ(CP)ǯPPAPBC

\PAPC=\QPBCǰ (CQ) (CP)ǰ\QCPB=\PACPǯ

PPACPQPB ǰ (CQ) (PAPB)ǯ

ę P2(R)ǰ

ǯ [x:

y:z] [a2yz:b2zx:c2xy]ǻaǰbǰc

Ȃ Ȃ Ȃęx+y+z= 0ǰ

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