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Benelux
2023 Benelux
3
3
Part of
2023 Benelux
Problems
(1)
Lines intersecting on the circumcircle (BxMO 2023, Problem 3)
Source: BxMO 2023, Problem 3
5/6/2023
Let
A
B
C
ABC
A
BC
be a triangle with incentre
I
I
I
and circumcircle
ω
\omega
ω
. Let
N
N
N
denote the second point of intersection of line
A
I
AI
A
I
and
ω
\omega
ω
. The line through
I
I
I
perpendicular to
A
I
AI
A
I
intersects line
B
C
BC
BC
, segment
[
A
B
]
[AB]
[
A
B
]
, and segment
[
A
C
]
[AC]
[
A
C
]
at the points
D
D
D
,
E
E
E
, and
F
F
F
, respectively. The circumcircle of triangle
A
E
F
AEF
A
EF
meets
ω
\omega
ω
again at
P
P
P
, and lines
P
N
PN
PN
and
B
C
BC
BC
intersect at
Q
Q
Q
. Prove that lines
I
Q
IQ
I
Q
and
D
N
DN
D
N
intersect on
ω
\omega
ω
.
BxMO
geometry