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International Contests
Lusophon Mathematical Olympiad
2016 Lusophon Mathematical Olympiad
2
2
Part of
2016 Lusophon Mathematical Olympiad
Problems
(1)
Center and midpoint
Source: Lusophon MO
2/17/2018
The circle
ω
1
\omega_1
ω
1
intersects the circle
ω
2
\omega_2
ω
2
in the points
A
A
A
and
B
B
B
, a tangent line to this circles intersects
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
in the points
E
E
E
and
F
F
F
respectively. Suppose that
A
A
A
is inside of the triangle
B
E
F
BEF
BEF
, let
H
H
H
be the orthocenter of
B
E
F
BEF
BEF
and
M
M
M
is the midpoint of
B
H
BH
B
H
. Prove that the centers of the circles
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
and the point
M
M
M
are collinears.
geometry