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2024-IMOC
G4
G4
Part of
2024-IMOC
Problems
(1)
APMO intersects on Luna Cabrera
Source: 2024 imocsl G4 (Boring Competiton P7)
8/8/2024
Given triangle
A
B
C
ABC
A
BC
with
A
B
<
A
C
AB<AC
A
B
<
A
C
and its circumcircle
Ω
\Omega
Ω
. Let
I
I
I
be the incenter of
A
B
C
ABC
A
BC
, and the feet from
I
I
I
to
B
C
BC
BC
is
D
D
D
. The circle with center
A
A
A
and radius
A
I
AI
A
I
intersects
Ω
\Omega
Ω
at
E
E
E
and
F
F
F
.
P
P
P
is a point on
E
F
EF
EF
such that
D
P
DP
D
P
is parallel to
A
I
AI
A
I
. Prove that
A
P
AP
A
P
and
M
I
MI
M
I
intersects on
Ω
\Omega
Ω
where
M
M
M
is the midpoint of arc
B
A
C
BAC
B
A
C
. [hide = Remark] In the test, the incenter called
O
O
O
and the circumcircle called
L
u
n
a
Luna
Lu
na
C
a
b
r
e
r
a
Cabrera
C
ab
rer
a
You have to prove
A
P
∩
M
O
∈
L
u
n
a
AP \cap MO \in Luna
A
P
∩
MO
∈
Lu
na
C
a
b
r
e
r
a
Cabrera
C
ab
rer
a
Proposed by BlessingOfHeaven
geometry
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