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Junior Balkan MO
2002 Junior Balkan MO
2
2
Part of
2002 Junior Balkan MO
Problems
(1)
the centers of the circles are on opposite sides of AB
Source: 6th JBMO 2002, Problem 2
6/19/2004
Two circles with centers
O
1
O_{1}
O
1
and
O
2
O_{2}
O
2
meet at two points
A
A
A
and
B
B
B
such that the centers of the circles are on opposite sides of the line
A
B
AB
A
B
. The lines
B
O
1
BO_{1}
B
O
1
and
B
O
2
BO_{2}
B
O
2
meet their respective circles again at
B
1
B_{1}
B
1
and
B
2
B_{2}
B
2
. Let
M
M
M
be the midpoint of
B
1
B
2
B_{1}B_{2}
B
1
B
2
. Let
M
1
M_{1}
M
1
,
M
2
M_{2}
M
2
be points on the circles of centers
O
1
O_{1}
O
1
and
O
2
O_{2}
O
2
respectively, such that
∠
A
O
1
M
1
=
∠
A
O
2
M
2
\angle AO_{1}M_{1}= \angle AO_{2}M_{2}
∠
A
O
1
M
1
=
∠
A
O
2
M
2
, and
B
1
B_{1}
B
1
lies on the minor arc
A
M
1
AM_{1}
A
M
1
while
B
B
B
lies on the minor arc
A
M
2
AM_{2}
A
M
2
. Show that
∠
M
M
1
B
=
∠
M
M
2
B
\angle MM_{1}B = \angle MM_{2}B
∠
M
M
1
B
=
∠
M
M
2
B
. Ciprus
geometry
trapezoid