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Problems
Contests
National and Regional Contests
Mexico Contests
Mexico National Olympiad
2023 Mexico National Olympiad
5
5
Part of
2023 Mexico National Olympiad
Problems
(1)
Two circles meet on AB
Source: Mexico 2023/5
11/8/2023
Let
A
B
C
ABC
A
BC
be an acute triangle,
Γ
\Gamma
Γ
is its circumcircle and
O
O
O
is its circumcenter. Let
F
F
F
be the point on
A
C
AC
A
C
such that the
∠
C
O
F
=
∠
A
C
B
\angle COF=\angle ACB
∠
COF
=
∠
A
CB
, such that
F
F
F
and
B
B
B
lie in opposite sides with respect to
C
O
CO
CO
. The line
F
O
FO
FO
cuts
B
C
BC
BC
at
G
G
G
. The line parallel to
B
C
BC
BC
through
A
A
A
intersects
Γ
\Gamma
Γ
again at
M
M
M
. The lines
C
O
CO
CO
and
M
G
MG
MG
meet at
K
K
K
. Show that the circumcircles of the triangles
B
G
K
BGK
BG
K
and
A
O
K
AOK
A
O
K
meet on
A
B
AB
A
B
.
geometry