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2021-IMOC
G3
G3
Part of
2021-IMOC
Problems
(1)
Incenter geo, prove that AR is parallel to BC
Source: IMOC 2021 G3
8/11/2021
Let
I
I
I
be the incenter of the acute triangle
△
A
B
C
\triangle ABC
△
A
BC
, and
B
I
BI
B
I
,
C
I
CI
C
I
intersect the altitude of
△
A
B
C
\triangle ABC
△
A
BC
through
A
A
A
at
U
U
U
,
V
V
V
, respectively. The circle with
A
I
AI
A
I
as a diameter intersects
⊙
(
A
B
C
)
\odot(ABC)
⊙
(
A
BC
)
again at
T
T
T
, and
⊙
(
T
U
V
)
\odot(TUV)
⊙
(
T
U
V
)
intersects the segment
B
C
BC
BC
and
⊙
(
A
B
C
)
\odot(ABC)
⊙
(
A
BC
)
at
P
P
P
,
Q
Q
Q
, respectively. Let
R
R
R
be another intersection of
P
Q
PQ
PQ
and
⊙
(
A
B
C
)
\odot(ABC)
⊙
(
A
BC
)
. Show that
A
R
∥
B
C
AR\parallel BC
A
R
∥
BC
.
geometry
incenter