Corrigé - Animath

A, B, C, D A0C0
A C (BD)B0D0B
D(AC)A0, B0, C0, D0
A
B
C
D
A0
C0
B0
D0
A, B, A0, B0[AB]
(BA0, BA) (B0A0, B0A) (BD, BA) = (B0A0, B0D0)
(CD, CA)=(C0A0, C0D0) (CD, CA)=(BD, BA)A, B, C, D
(B0A0, B0D0) = (C0A0, C0D0)A0, B0, C0, D0
[BD]ABCD E O1
O2ABE ADE
AO1EO2
A
B
CD
E
O1
O2
\
AO1E= 2
\
ABE = 90◦O1A=O1E AO1E
O1AO2E O2
O1O2(AE)
AO1EO2
ABCD
K A, B, K (BC) (AD)M
N KM =KN

A
B
C
D
K
M
N
(NM, N K)=(BM, BK) = (BC, BD) = (AC, AD) = (AK, AN ) = (MK, MN)
MKN K
ABC b B
π/3 [AD] [CE]H
ABC
\
AHE
\
CHD
A
B
C
D
E
H
O
B, E, H, D [BH]
\
DHE = 180◦−
\
EBD =
120◦
\
AHC = 120◦
[
AOC = 2
\
ABC = 120◦A, C, O, H
\
CHO =
[
CAO =1
2(
[
CAO +
[
OCA) = 1
2(180◦−
[
AOC) = 90◦−
\
ABC =
30◦
(CH)⊥(AB) (AH)⊥(BC)
\
CHD =
\
ABC = 60◦O
\
CHD
ABC X Y
AXB [AB]
BY C [BC]
\
AXB +
\
BY C = 180◦
Z[AC] (XZ) (Y Z)
D E [BC] [AB]

A
B
C
D
E
X
Z
Y
•(XE)⊥(DZ) (DZ)// (AB)
•(EZ)⊥(DY ) (EZ)// (BC)
•XE
DZ =XE EA =CD DY =EZ DY
XEA CDY
XEZ ZDY (XE)⊥(DZ) (XZ)⊥
(ZY )
B
\
ABC ω
[BA) [BC)K
ω[BA)
\
ABC K
P(BC)M[P M ]
ω
\
ABC
A
B
C
K
∆
MP
O
N
∆0
O ω ∆
\
ABC ∆0∆
O N O ∆
s∆0∆0O O P K (P K)⊥∆0∆0
O
s∆∆O N K M
s∆◦s∆0O N P M s∆◦s0
∆
MN OP P M =ON = 2d(O, ∆)

ABCD
F1F2
[AC] [BD]
ABCD F1F2
ABCD
I, J, K, L, M, N [AC],[BD],[AB],[BC],[CD] [DA]
A
B
C
D
I
J
K
L
M
N
(IK) (JM) (BC)
(IM)// (JK)IMJK INJL
[~u, ~v]~u ~v
XY ZT −−→ XY =~u −−→ Y Z =~v +XY ZT
−
•[~u, ~v] = −[~v, ~u]
•[~u, ~v +~w] = [~u, ~v]+[~u, ~w]
•[~v +~w, ~u] = [~v, ~u]+[~w, ~u]
P
P1P2P P1P2
IK =BC/2KJ =AD/2IMJK ±1
4[−−→
AD, −−→ BC]
INJM ±1
4[−−→
AB, −−→ CD]F1F2
[−−→ AB, −−→ CD] = ±[−−→ AD, −−→ BC].
[−−→ AB, −−→ CD]=[−−→ AD, −−→ BC]
⇐⇒ [−−→ AB, −−→ AD]−[−−→ AB, −→ AC] = [−−→ AD, −→ AC]−[−−→ AD, −−→ AB]
⇐⇒ [−→ AC, −−→ AB]=[−−→ AD, −→ AC]
⇐⇒ (AC)

[−−→ AB, −−→ CD] = −[−−→ AD, −−→ BC] (BD)
IMJK INJL
(AD)// (BC)IMJK INJL
(AB)// (CD)
ABCD
(AD) (BC)
(AB) (CD)
X= [DA)∩[CB)Y= [BA)∩[CD)
A
B
C
D
I
J
K
L
M
N
X
Y
IMJK
KJ ·KI sin
[
IKJ =1
4BC ·AD sin
\
AXB.
INJL
NJ ·N I sin
[
JNI =1
4AB ·CD sin
\
AY D.
AXB AY D
sin
\
AXB
sin
\
AY D =AB sin
\
XAB
BX
DY
AD sin
\
Y AD =AB ·DY
BX ·AD .
IMJK INJL
CB
BX =CD DY .
(XY )// (BD)
(AC)ABCD J
(AC)AJB AJD CJB
CJD
XBDY C =
(XB)∩(DY )A= (XD)∩(BY ) [BD] (XY )
(BD)
(XY ) (BD)C
B D X Y [BD]
[XY ]C A
(AC)U[BD]Y0
(CY ) (BD)X A0= (BY 0)∩(DX)