Cours - Géométrie par Pierre DEhornoD Table des matières 1 Les

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¡=“Bh•yOQ•l=
—
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fkute
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u ute_ fkuwr&ikfki\ux‰‡ŠGu ;
u_ fkuwr&ikf\ikut‰-ŠGu 221 Z
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1 221 Z
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y
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vxj vtq vxf\i!“‰Guxeeiut_7eux‰Gf\uxr&uxdy_7ecijld3^ 1  2 ‡ 1 5 2 Neci  ut_I5 ecj[d“_7ŠNu1v[_bste7jRjlecste
^[aab^ Rjla _ -fk^‡ŠGajli\_c<
u 7 12 8 R^[fkjlae 1 2 % u_"5 ejldy_ vtj›vq›vtfki ‰Nuxe eiu_ ecut‰Gfkutr‡utdy_
eci jld ^ 1  2 ‡ — ‚ 1 5 2 Z $ ~u Gf\‰Gemutd ^Ruxfk^[dy_ f\u‡vtuxdy_bauŠG‰ vtuxavxfku‡vxi\acvxj[dGecvtaci\_
1 2 ?5WRj[d^ 7 ” 1 S” 2 8v‡3>7' 1  2 8.‡3%7Q5 1 J5 2 8 Z
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u BCl‰Ga
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i
A
B
A
R
C
B
D
P
M
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Lestriangles
M AB etM B C
sont inversemen
t semblables.
‚
A
Théor̀eme d
’“JRB M*=(X O TI= W
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A
B
Q
R
T
C
B
D
C
P
Théor̀eme de Miquel
ƒ< V*X]9V 5
y jliˆ_~}œf ƒ~i\d“_cuxaecuxv_bi\jld–ŠGuxe3vtuxacvtfkute vtikavxjldGevxai\_be3^h‰› _cacig^hdlfkute
M
riangles
M AB etM B C
inversemen
t semblables.
vxjlf\ikdGsx^[aciˆ_bs ŠNuxe, jli\dy_bejld ^
} ‡N— ‚ 1 # } } 1
Œ^[a7vxj vtq vtfkikvti\_cs jld ^
A
}
2
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A
1
1
ut_
2 ;
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B
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1
O
D
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vxj vtq vxf\!i “‰Guxe3 ŠGjldGv#} ^ ^[a _bi\uxdy_^[‰ vxuxavxf\u1vtikavxjldGevxai\_^[‰6_baig^[dlfku 1 ;&Z
f uxe _ i\@r Rjla_b^[dy_1fkjlae“‰ ƒ…jld*vx^[fkvt‰Gfku&ŠGuxe^[dlf\uxemŠGu@+Q^[i\acu&^h__buxdy_cikjld*^[‰› ecildGuxemut_
f ƒ…jlaci\uxdy_|^h_cikjld ŠGuxe1^[d lfkutew"^ B d ŠGuAdGu ^hewsxvxaikau`ŠGuxenv|„Gjleuxe ab^[i\uxewec‰Nawvxuxa _|^[i\dGuxe#BC[‰Gacute
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“‰Gi
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^ BCl‰Nacu “‰ ƒ~j[d(^AŠGuxeecikdNsxu Rut_jl2d lstaci BGu “‰ ƒ…uxf\fkute auxe_cuxdy,_ ›ac^[ikute e‰Ga7fkute
^[‰N_baux&e BCl‰Nacux3e jl‰ ^[‰3r‡jli\dG&e “‰GunŠGuxeacutfg^h_cikjldNeeci\r‡i\fg^[i\acuxe acute_cuxdyA_ ac^[ikutexZ
Ёd^h‰N_bau1asxe‰Gf\_b^h_7ikr j[a_|^hd“_ ec‰Ga7f\uxe7vxutacvxf\uxe7ut_f\ux&e “‰ ^[ŠGaikfk^h_bŽtacuxeute_ fkuwe‰Gi h^[dy_CZ
’“JLB ›M = ™ n@PJRW”’›M ’ = P
wd
C
r
E
C
B
D
Théor̀eme de Ptoĺ
emée
Puissance
de P parrapportà :
P A.PB = P C .PD = P E 2 = O P 2 − r2
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1221 +h% ).+X,X/K" E ,.'&/?# 9! % , E#" "5%$
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D
M
Théor̀eme de Mique
Lestriangles
M AB etM B C
sont inversemen
t semblables.
A
B
O
P
C
r
E
E
C
B
Puissance
de P parra
P A.PB = P C .PD =
D
ƒ< V*X]9V 5
Théor̀eme de Ptoĺ
emée
y jli\_–f ƒ…‰GdGi“‰Gu. jli\dy_9ŠG‰Gfk^[dw_butf]“‰Gu fkute_bacik^[dlf\uxe 1 2 –ut_
ejlikutd“_IŠGi\acutvt_butr‡utdy_
ecuxrGfg^[Nfkuxe&FQv[ƒ…uxe _>G ŠGi\acu&“‰ ƒ…ikf›u›ike _bu7‰GdGu7eci\r‡i\fkiˆ_b‰GŠGu>ŠGi\acutvt_buutdEljPqy^[dy_ 12 e‰Ga 1 1 IZ
wd ^ 2 1 ‡ 12 1 Š ƒ…jb 2 ‡ 1 2 1 1 Z $mƒ{^[‰›_bacuL ^ha_Ajld ^ 11 ‡
12 ‡ 1.1 1 6 ŠGj[dGv fkute‡_baig^[dlfkute 11
2 1 u_ 1
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1
) 12 ‰ 2 "^Jluxv1sl^[fkiˆ_bs eiut_ecut‰Gfkutr‡utd“_eifkuxe&Rjli\d“_ce 1 2 ecj[d“_ ^[f\!i ldGsteŠ ^[dNe
vxut_ jlacŠGau[Z 4 d acuxrGfk^jC^[dy_ 2 8u_ 1 ^[a fkuteAh^[fkut‰Gace jlN_cuxd“‰Gux3e jld _cacjla‰ lu f ƒ~i\dGs y^hfki\_cs
ŠGumf ƒ~stdGjldGvts[Z ] ƒ~s3y^[fkiˆ_bsn^&fkikut‰ ei9ut_ eux‰Gf\uxr&uxdy_ eciIjld(^ 1221 ‡`— ‚ 12 ‡`— ‚ 1 1 ŠGjldGvneciu_ecux‰Nfkuxr&uxdy_ecif\uxe&Rjlikdy_ce 1 2 1 ejldy_vxj vtq vxf\!i “‰GuxeŠ ^[dNevtut_jlaŠGacuhZ ЁdGu`^[‰›_bacu GacjGaci\st_bsmikr j[a_|^hd“_cu “‰GiIŠGstvxjl‰Gf\u ŠGu fk@^ Gauxr&ikŽtacu1uxe_ vx
u “‰ ƒ…jld ^ Ruxf\fku
fg^
G‰Gi\ecec^[dGvxu1Š ƒ…‰G2d jli\dy&_ ^[a7ab"^ j[a,_ `‰Gd vxuxavxf\u[Z
’“JLlB M*=  mTIO ?C?tD F•l=S ˆTIF U JRO F @&UIDGBB€DGUIU JLB€
@ TF–•l=“BE•yW”= 1 1
A
B
!" ' +H% @B#/1@E # #G' +H% 24!"5%*' ; 9 !" ' +H%*# </1!" 'R#2 ,h6D6,h%('i2 ,h/
#% 1 #' 2 - G#%('K+<#E E #I #%('4@B!% !G%$(+H64 9 E !/K6=E #F2A/1! (R +h" ' ; 1 ;
#' # 7 2 ,h6 # E , X/?!" 'f# 9
A
;
#' @ !+g2 ,h%(' E$#@B#/1@E #
*% Q# D
R2=#&%$ ).+# # ;
2
T
D estl’axe
radical
de
etG .
ƒ< V*X]9V 5
y jli\_‰GdGum^[‰N_cacu1ŠGacj[i\_bu ^heceb^hd“_A ^[a-; tu _ vxj[‰ ^[dy_ fku1vxuxavxf\umutd 1 ut_ F=ljli\a&BCl‰Gau7IZ
wd ^
; 11 ‡ 2 11 ‡ 2 1 ‡
‚ ;- 2 9
]
t
u
e
b
_
a
g
i
[
^
[fkuxe ; 11 u_ ; 2 ecj[d“_7ŠNjldGv euxr Gfk^[Gf\uxeF”Théor̀
d
ut_Š ƒ…jleme
aikuxdyde_b^hMiquel
_bi\jldGe j j[ecsxuteSI GŠ ƒ…jb
Lestriangles
; 2 B GCecjliˆ_I; 1 ; 2 ‡ ; 1 ;Z
; 1 ;- M‡ AB
; 1etM
sont inversemen
t semblables.
1
C
B
D
2
C
P
M
]u(G aj›ŠN‰Gi\_2; 1 ; 2 uxe _&^Ruxf\s
X% a;!USUT[|PSVVŠGu ; ^[a ab^ jla _ ^[‰
vxuxavxf\u: Z y i” uxe _ f\u vtuxdy_bau ŠG‰
vxuxavxf\uut_'& ejldAab^Cqljld|yjld ut‰N_>v|„Gjli ecika 7 ” ;8vtjlr&r‡u>ŠGajli\_cu u_u aGaci\r‡uta
fg^
G ‰Gi\ecec^[dGvxu1ŠGu; vtjlr‡r&u
7 ” ; ‰ & 87 ” ; ‚ & 8 ‡Q” ; ( ‚ & ( A
A
E
C
B
D
Théor̀eme de Ptoĺ
emée
B
O
D
P
C
r
‹
E
&S ] ^ b a
(
( !$#
"
(
Puissance
de P parrapportà :
P A.PB = P C .PD = P E 2 = O P 2 − r2
M
Théor̀eme de Miquel
Lestriangles
M AB etM B C
sont inversemen
t semblables.
A
>d lstdGsxac^[fGjld`‰›_bikf\ikeu ŠGute f\jldl‰Nux‰Gae
^[!f lstGaci“‰Guxe3 vtu• ‰Niw^Rjl‰GavtjldGecs ‰NuxdGvtu“‰Gufk^•G‰Gikeeb^[dNvxuute_@ j[eci4
_bi [u ei ; uxe&_ f ƒ…u _bstaci\ux‰Ga ŠG"u ut_dGsy^E_bi[u ei ; uxe_&f ƒ…ikdy_bstaci\ux‰GatZ›]u vC^[ef\ikr&i\_cu 1 ‡ 2
vxjlaacux>e j[dGŠ(^[‰3vx^[e j b fk^AŠNacjliˆ_bu 7 ; 1 8 ute__|^[dluxdy_bum^[‰ vxutacvxf\uC ^haf\u _b„GstjlacŽtr‡u1ŠGu1ŒqE
_b„ ^ [jlac"u jld aut_cacjl‰ [umŠNikacutvt_cuxr&uxdy_ fk
^ [^hfkux‰NaŠGumfk^ G‰Gikeeb^[dNvxuutd2+”j[dGvt_cikjld ŠGu” ; u_ ŠNu
&›Z
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ŠGacjliˆ_bute 7 12 8>ut_ 7 1 8 ecu vxjl‰ Ruxdy_utd‰G(d Rjlikdy_ Œ ut_ “‰ ƒ…jld3^ ; 1 ; 2 ‡?; 1 ;- F ^Jlutv
ŠGuxe7fkj[d l‰Gut‰Gace^[f lsxGa!i “‰GutSe I*G^[fkjlae 1 2 1 ut_ ecjldy_vxj vtq vtfk!i “‰GutexZ
y j[ikuxdy_ ŠGut‰› vxutacvtfkuxe ut_ (
ŠNu vtuxdy_bauxe-aux>e utvt_b4i +”e ” ut@_ ”(-ŠNu
ac^CqljldGeIacute Ruxv_bi +”e & ut_ &(JP‰GdG1u “‰Guxe _cikjld8d ^h_c‰Gacutfkfku uxe _–ŠGu eu ŠGuxr‡^[ad ŠNuxa ‰Nuxf>uxe _ f ƒ…uxdNecuxrGfkuAŠGute Rjlikdy_be
^Cql^[dy_ r ?xr&&u G‰Gi\ecec^[dGvx,u ^[a ac^ Rjla_
vtuxe7ŠGut‰› vxutacvxf\uxet„Z d ec^[i\_ [a [vtu dNj[_bau&dNj[_bau +”j[acr ‰Gf\u`ua Gf\ikvxiˆ_b
u “‰ ƒ…ikf
etƒ{"^ li\_&ŠGu f ƒ…uxdG( euxr Gf\( u ŠNux~e Rjl( ikdy_c&e ; (
[sxai BL^[dy4
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_c„Gsxj[acŽxr&uw‚ ŠGu1Œq“_b„ ^ [jlacun‚ jld3r&jldy_bacu
“‰Gu*f ƒ…uxdGeuxr Gf\u*ŠGute(_butfk
e Rjli\d“_c9e ;
a NORP=Y T XNpQmT -S 1TBY XNO XN N&Q &YZNO
ute_ ‰GdGu ŠGajli\_cNu Rux>a utdGŠGi\vx‰Gfk^[ikaOu f ƒ{^€N.u 7 ””('8^ utfks TJYaV<>T ;'PT^CŠGute
ŠGux‰›vxutacvtfkuxe ut"_ (PZ y iIŠGux‰›vxutacvtfkuxe ecu vtjl‰ Ruxdy_wuxdŠGux‰› j[ikdy_be 1 u_ 2 ^[fkjlae f\ux‰Ga
^E›u-ab^hŠGikvx^[f uxe _ fg^3ŠGajli\_c9u 7 12 8 Z y i ŠGut‰›*vxuxavxf\uxeejldy__b^[d lutd“_ceutd ‰Gd Rjli\d“_ 1 I^[f\jlace
fkux‰Na7^E›u1ac^[ŠGi\vC^[f ute_fg^_b^[d lutdy_bunvxjlr&r ‰GdNu “‰Gif\uxees ^[au[Z
d ^`fku _b„NsxjlaŽxr&uwec‰G4i [^hd“_e‰Ga7fkuxe7^€Nute ac^[ŠGi\vC^[‰› Z
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A
B
O
D
P
C
r
E
E
C
B
Puissance
de P parrapportà :
P A.PB = P C .PD = P E 2 = O P 2 − r2
D
Théor̀eme de Ptoĺ
emée
D
D estl’axe
radical
de
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