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Greece Contests
Greece National Olympiad
2024 Greece National Olympiad
2
2
Part of
2024 Greece National Olympiad
Problems
(1)
Geometry with circle centered at arc midpoint
Source: Greece National Olympiad 2024, Problem 2
2/24/2024
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
<
A
C
<
B
C
AB<AC<BC
A
B
<
A
C
<
BC
with circumcircle
Γ
1
\Gamma_1
Γ
1
. The circle
Γ
2
\Gamma_2
Γ
2
has center
D
D
D
lying on
Γ
1
\Gamma_1
Γ
1
and touches
B
C
BC
BC
at
E
E
E
and the extension of
A
B
AB
A
B
at
F
F
F
. Let
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
meet at
K
,
G
K, G
K
,
G
and the line
K
G
KG
K
G
meets
E
F
EF
EF
and
C
D
CD
C
D
at
M
,
N
M, N
M
,
N
. Show that
B
C
N
M
BCNM
BCNM
is cyclic.
geometry