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Benelux
2024 Benelux
3
3
Part of
2024 Benelux
Problems
(1)
2024 BxMO P3
Source: 2024 BxMO P3
4/28/2024
Let
A
B
C
ABC
A
BC
be a triangle with incentre
I
I
I
and circumcircle
Ω
\Omega
Ω
such that
∣
A
C
∣
≠
∣
B
C
∣
\left|AC\right|\neq\left|BC\right|
∣
A
C
∣
=
∣
BC
∣
. The internal angle bisector of
∠
C
A
B
\angle CAB
∠
C
A
B
intersects side
B
C
BC
BC
at
D
D
D
and the external angle bisectors of
∠
A
B
C
\angle ABC
∠
A
BC
and
∠
B
C
A
\angle BCA
∠
BC
A
intersect
Ω
\Omega
Ω
at
E
E
E
and
F
F
F
respectively. Let
G
G
G
be the intersection of lines
A
E
AE
A
E
and
F
I
FI
F
I
and let
Γ
\Gamma
Γ
be the circumcircle of triangle
B
D
I
BDI
B
D
I
. Show that
E
E
E
lies on
Γ
\Gamma
Γ
if and only if
G
G
G
lies on
Γ
\Gamma
Γ
.
geometry