Feuille 21 : Géométrie : Barycentre. 1) Soit ABC un triangle. . . c

BCP ST 1B 
ABC
G{(A, 1),(B, 2),(C, 3)}
G0{(A, 1),(B, 3),(C, −3)}
(AG0) (BC)
(O,~ı,~,~ k)
A(1,0,2) B(−2,−1,3) C(0,0,1) D(−1,2,0)
ABCD
(O,~ı,~)
A(1,2) B(−3,4) C(−2,5)
G(A, 3) (B, 2) (C, −4)
(BG)
ABC A0{(B, 2),(C, −3)}
B0{(A, 5),(C, −3)}C0{(A, 5),(B, 2)}
G{(A, 5),(B, 2),(C, −3)}(AA0) (BB0)
(CC0)
ABC
ΓM
−−→
MA +−−→ MB + 2 −−→ MC 
=
−−→
MB + 3 −−→ MC 
ABCD G {(A, 4),(B, 1),(C, 1),(D, 1)}
H B C D
G(AH)
G(AH)
A1A2A3(λ, µ)∈R2λ+µ6= 0 G1=bar{(A2, λ),(A3, µ)}
G2=bar{(A3, λ),(A1, µ)}G3=bar{(A1, λ),(A2, µ)}
A1A2A3G1G2G3
ABC G {(A, 2),(B, −1),(C, 1)}
E M 
2−−→
MA −−−→ MB +−−→ MC 
=AB
E
ABC k I J K
−→
BI =k−−→ BC −→ CJ =k−→ CA −−→ AK =k−−→ AB
k=1
3
k G I J K

ABCD K (A, 2) (B, −1) (C, 2)
(D, 1)
I(A, 2) (B, −1) J(C, 2) (D, 1)
I J
2−−→ KA −−−→ KB 2−−→ KC +−−→ KD
K(I, 1) (J, 3)
K
[AB]
M MA = 2MB
MA = 2MB ⇐⇒ −−→ MA −2−−→ MB .−−→ MA + 2 −−→ MB = 0
ABC a =BC b =AC c =AB
I{(A, a),(B, b),(C, c)}
(O;~ı,~)
A(1,2) B(3,0) M0(0,0)
n Mn
n Mn+1 A B Mn
A, B, M0, M1, M2
Mn
n∈N(xn, yn)Mn
xn+1 yn+1 xnyn
xnynn
(Mn)C
ABCD
•G{(A, 1),(B, 2),(C, 3),(D, 3)}
•G1{(A, 1),(B, 2)}
•I[CD]
A, B, C, D, G1I G1
G{(G1,1),(I, 2)}
G
(A, −−→ AB , −−→ AD ) (a, b)C
G(A, −−→ AB , −−→ AD )
a b A, G C
A, G C