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Turkey Contests
Turkey MO (2nd round)
2023 Turkey MO (2nd round)
2
2
Part of
2023 Turkey MO (2nd round)
Problems
(1)
Shouting inversion but has an elegant solution with La Hire
Source: 2023 Turkey NMO 2nd Round P2
12/21/2023
Let
A
B
C
ABC
A
BC
be a triangle and
P
P
P
be an interior point. Let
ω
A
\omega_A
ω
A
be the circle that is tangent to the circumcircle of
B
P
C
BPC
BPC
at
P
P
P
internally and tangent to the circumcircle of
A
B
C
ABC
A
BC
at
A
1
A_1
A
1
internally and let
Γ
A
\Gamma_A
Γ
A
be the circle that is tangent to the circumcircle of
B
P
C
BPC
BPC
at
P
P
P
externally and tangent to the circumcircle of
A
B
C
ABC
A
BC
at
A
2
A_2
A
2
internally. Define
B
1
B_1
B
1
,
B
2
B_2
B
2
,
C
1
C_1
C
1
,
C
2
C_2
C
2
analogously. Let
O
O
O
be the circumcentre of
A
B
C
ABC
A
BC
. Prove that the lines
A
1
A
2
A_1A_2
A
1
A
2
,
B
1
B
2
B_1B_2
B
1
B
2
,
C
1
C
2
C_1C_2
C
1
C
2
and
O
P
OP
OP
are concurrent.
geometry
circumcirle