Géométrie euclidienne en dimension 2 Exercice 1. Exercice 2

#
Cx2+y2−4x−2y= 0
CO(−2; 3) C
D x + 2y+ 4 = 0 A(−1; −1) B(3; 1)
A B D
CO1,r1CO2,r2CO3,r3D1, D2D3
C1∩ C2⊂D3C1∩ C3⊂D2C2∩ C3⊂D1
D3⊥(O1O2)
M∈D3⇔−−−→ O1M.−−−→ O1O2=1
2(O2O2
1+r2 1−r2 2)
D1D2D3
Dλ(1 −λ2)x+ 2λy + (λ2−2λ−3) = 0 λ∈R
E={M∈R2,∃λ / M ∈Dλ}=S
λ∈R
Dλ
{M∈R2,∃(λ, µ)∈R2/ M ∈Dλ∩DµDλ⊥Dµ}
ABC p I r
BC.−→ IA +CA.−→ IB +AB.−→ IC =−→ 0
ABC r p
ABC M P Q R M
(BC) (CA) (AB)M(AP ) (BQ)
(CR)
Cω r P A B C
P
P A.P B
Cx2+y2−2ax −2by +c= 0
P A.P B =x2
0+y2 0−2ax0−2by0+c
P(x0, y0)
A(0, a)d y =mx ∆n(Ox)M d
M0

[MM0]C
∆AC
ABC a
M(ABC)M A2+a2= 2(M B2+M C2)
ABC CM
(x, y, z) (ABC)
M∈ C ⇔ xAM2+yBM2+zCM2= 0 ⇔xyAB2+xzAC2+yzBC2= 0
C=C(O, r)f f(C) = Cf
O
λ∈RDλ(1 −λ2)x+ 2λy = 4λ+ 2
ΩDλ
D, D0O
H={M d(M, D) = d(M, D0)}
H(D, D0)
s s(D) = D0s
HCDCD D0
H
F1, F2, F3F
F1, F2, F3
ABC α, β, γ
ρ, ρ0, ρ00 A, B, C α, β, γ
ρ◦ρ0◦ρ00
G
G
f, g ∈G f g f ◦g◦f−1◦g−1
G
G p
G
G
E O A
(a, 0) M M0=f(M)A, M, M0
(MO) (M0O)f(x, y)M
f[AO]
x < 0, y > 0

Ω(2; 1) r=√5
x2+y2+ 4x−6y+ 8 = 0 x2+y2+ 4x−6y−32 = 0
Ω1(1; 0) r1=√2 Ω2(−7
8;15 4)r2=17 8√5
{A, B}=C1∩ C2(O1O2) [AB]
−−−→
O1O2.−−→ AM = 0 ⇔−−−→ O1M.−−−→ O1O2=−−−→ AO1−−−−→ O1O2r2
2= (−−→
AO1+−−−→ O1O2)2⇒
−−−→
O1M.−−−→ O1O2=1
2(−−−→
O2O1
2+r2
1−r2 2)
2d2D3M∈D2∩D3−−−→ O2M.−−−→ O2O3=1
2(O2O2
3+r2 2−r2 3)
M∈D1
x0= 1 y06= 1 x06= 1 2 λ∆=(x0−2)2+
(y0−1)2−1≥0MinE M Ω(2; 1)
1
x06= 1 −→ nλ(1 −λ2,2λ)−→ nµ(1 −µ2,2µ)
−→
nλ.−→ nµ= 0 ⇔(1 −λ2)(1 −µ2)+4λµ = 0 λ+µ= 2 1−y0
1−x0
λµ =x0−3
1−x0
(λµ)2−(λ+µ)2+ 6λµ + 1 = 0 ⇔x2
0+y2 0−4x0−2y0+ 3 = 0 ⇔(x0−2)2+ (y0−1)2= 2
(2,1) √2 (1,0) (1,2)
G{(A;a),(B;b),(C;c)}dC=d(G, (AB))
dC=(GAB)
c=|det(−→
GA, −−→ GB)|
2c=|det(−b−−→
GB −c−−→ GC, −−→ GB)|
2ac
=|det(−−→
GC, −−→ GB)|
2a=(GBC a=dA.
dA=dB=dCG=I
M−a
n,0M0a m−n;ma m−n
x2+y2+xa(m−2n)
n(m−n)−yam m−n=a2 n(m−n)
Γx2+y2−2αx−2βy +γ= 0 Γ ⊥ C
a1
m−n−1 nα+am m−nβ=−a2 n(m−n)+γ
n

(γ= 0
2aα +amβ −mγ = 0
−amα +a2= 0 ⇔
α=a
m
β=−2a
m2
γ= 0
Γx2+y2−2a
mx+4a m2y= 0 .
A0BC ABC (BC)
xAM2+yBM2+zCM2=r2−OM2C=C(O, r)
xyAB2+xzAC2+yzBC2=xAM2+yBM2+zCM2
Ω = (1,2)
α+β+γ=π K (AC)
x0=ay2
x2+y2−ax y0=−axy x2+y2−ax M0M
[AO]
D[AO]D x2+y2−ax < 0
y > 0x0<0y0>0
D f