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Problems
Contests
National and Regional Contests
Mexico Contests
Mexico National Olympiad
2004 Mexico National Olympiad
5
5
Part of
2004 Mexico National Olympiad
Problems
(1)
intersection of two lines is the circumcirce of a triangle
Source: Mexican Mathematical Olympiad 2004 OMM P5
7/31/2018
Let
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
be two circles such that the center
O
O
O
of
ω
2
\omega_2
ω
2
lies in
ω
1
\omega_1
ω
1
. Let
C
C
C
and
D
D
D
be the two intersection points of the circles. Let
A
A
A
be a point on
ω
1
\omega_1
ω
1
and let
B
B
B
be a point on
ω
2
\omega_2
ω
2
such that
A
C
AC
A
C
is tangent to
ω
2
\omega_2
ω
2
in C and BC is tangent to
ω
1
\omega_1
ω
1
in
C
C
C
. The line segment
A
B
AB
A
B
meets
ω
2
\omega_2
ω
2
again in
E
E
E
and also meets
ω
1
\omega_1
ω
1
again in F. The line
C
E
CE
CE
meets
ω
1
\omega_1
ω
1
again in
G
G
G
and the line
C
F
CF
CF
meets the line
G
D
GD
G
D
in
H
H
H
. Prove that the intersection point of
G
O
GO
GO
and
E
H
EH
E
H
is the center of the circumcircle of the triangle
D
E
F
DEF
D
EF
.
geometry
circumcircle