Géométrie élémentaire, L2 maths-info

P
AB A B
P
A, B C A 6=C AB/AC
A B
PC D
C D A B
[A, D] [B, C]
(A, B) (C, D)−−→ AB =−−→ CD (A, B)
(C, D)
ERE
(x, y)E2x y xRy x
yR< > =≤ ≥ R
Rx E xRx
Rx y E xRy yRx
Rx y z E xRy yRz xRz
R< > =≤ ≥ =
R∗'x'y x/y > 0
'R∗

∼E
x∈E x Cl(x) = {y∈E, y ∼x}
∼E E
E x
ER
x∼x x ∈Cl(x)Cl(x)6=Cl(y)
Cl(x)∩Cl(y) = ∅z Cl(x)∩Cl(y)
Cl(x) = Cl(y)x0Cl(x)x0∼x
z Cl(x)z∼x x ∼z x0∼z
z∈Cl(y)z∼y x0∼y x0∈Cl(y)
Cl(x)⊂Cl(y)x∼y y ∼x x
y Cl(y)⊂Cl(x)Cl(x) = Cl(y)
=R
'R∗R>0R<0
P × P ∼ (A, B)∼(C, D) [A, D]
[B, C]ABDC
∼
P × P P
A B −→ v=−−→ AB
Cl((A, B)) = {(C, D)∈ P2,(C, D)∼(A, B)} ∼ P × P
(A, B)−→ v−→ P
−→
v−→ P(A, B)−→ v||−→ v||
A B −→ v−→ v
(C, D)−→ v ABDC
AB DC
−→
v−→ PAP
BP−→ v=−−→ AB A =B+−→ v

−→
v−→ w(A, B)
−→
v C P−→ w=−−→ BC
−→
v+−→ w=−→ AC.
(A, B)−→ v
(A0, B0)−→ v C0P
−→
w=−−→ B0C0ACC0A0ABB0A0
[AA0] [BB0]
BCC0D0[BB0] [CC0]
[AA0] [CC0]
ACC0A0
−→
u , −→ v−→ w−→ P−→ 0
−→
AA A P
(−→ u+−→ v) + −→ w=−→ u+ (−→ v+−→ w)
−→
0 + −→ v=−→ v+−→ 0 = −→ v
−→
u+−→ v=−→ v+−→ u
−→
v=−−→ AB −−→ v−−→ BA −→ v+(−−→ v) = −→ 0
(−→ P,+)
−→
v−→ PBP
AP−→ v=−−→ AB A =B−−→ v
B=A+−→ v
R
λ−→ v=−−→ AB
−→
v=−→ 0λ.−→ v=−→ 0
−→
v6= 0
λ≥0C[AB)AC/AB =λ λ.−→
v=−→ AC
λ≤0λ.−→ v= (−λ).(−−→ v)
(A, B)−→ v
λ µ R−→ v−→ w−→ P
λ.(µ.−→ v)=(λµ).−→ v
(λ+µ).−→ v=λ.−→ v+µ.−→ v λ.(−→ v+−→ w) = λ.−→ v+λ.−→ w
0.−→ v=λ.−→ 0 = −→ 0
(−→ P,+, .)R