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2016 USAMO
5
5
Part of
2016 USAMO
Problems
(1)
Angel Bisector and Equilateral Pentagon AMNPQ
Source: 2016 USAMO 5
4/20/2016
An equilateral pentagon
A
M
N
P
Q
AMNPQ
A
MNPQ
is inscribed in triangle
A
B
C
ABC
A
BC
such that
M
∈
A
B
‾
M\in\overline{AB}
M
∈
A
B
,
Q
∈
A
C
‾
Q\in\overline{AC}
Q
∈
A
C
, and
N
,
P
∈
B
C
‾
N,P\in\overline{BC}
N
,
P
∈
BC
. Let
S
S
S
be the intersection of
M
N
↔
\overleftrightarrow{MN}
MN
and
P
Q
↔
\overleftrightarrow{PQ}
PQ
. Denote by
ℓ
\ell
ℓ
the angle bisector of
∠
M
S
Q
\angle MSQ
∠
MSQ
.Prove that
O
I
‾
\overline{OI}
O
I
is parallel to
ℓ
\ell
ℓ
, where
O
O
O
is the circumcenter of triangle
A
B
C
ABC
A
BC
, and
I
I
I
is the incenter of triangle
A
B
C
ABC
A
BC
.
geometry