MathDB
Problems
Contests
National and Regional Contests
Cuba Contests
Cuba MO
2009 Cuba MO
6
6
Part of
2009 Cuba MO
Problems
(1)
collinear, intersecting circles related
Source: 2009 Cuba 2.6
8/27/2024
Let
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
be circles that intersect at points
A
A
A
and
B
B
B
and let
O
1
O_1
O
1
and
O
2
O_2
O
2
be their respective centers. We take
M
M
M
in
ω
1
\omega_1
ω
1
and
N
N
N
in
ω
2
\omega_2
ω
2
on the same side as
B
B
B
with respect to segment
O
1
O
2
O_1O_2
O
1
O
2
, such that
M
O
1
∥
B
O
2
MO_1\parallel BO_2
M
O
1
∥
B
O
2
and
B
O
1
∥
N
O
2
BO_1 \parallel NO_2
B
O
1
∥
N
O
2
. Draw the tangents to
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
through
M
M
M
and
N
N
N
respectively, which intersect at
K
K
K
. Show that
A
A
A
,
B
B
B
and
K
K
K
are collinear.
geometry
collinear