MathDB
Problems
Contests
National and Regional Contests
Moldova Contests
Moldova Team Selection Test
2012 Moldova Team Selection Test
7
7
Part of
2012 Moldova Team Selection Test
Problems
(1)
Externally tangent circles and midpoints
Source: Moldova TST 2012
2/19/2016
Let
C
(
O
1
)
,
C
(
O
2
)
C(O_1),C(O_2)
C
(
O
1
)
,
C
(
O
2
)
be two externally tangent circles at point
P
P
P
. A line
t
t
t
is tangent to
C
(
O
1
)
C(O_1)
C
(
O
1
)
in point
R
R
R
and intersects
C
(
O
2
)
C(O_2)
C
(
O
2
)
in points
A
,
B
A,B
A
,
B
such that
A
A
A
is closer to
R
R
R
than
B
B
B
is. The line
A
O
1
AO_1
A
O
1
intersects the perpendicular to
t
t
t
in
B
B
B
at point
C
C
C
, the line
P
C
PC
PC
intersects
A
B
AB
A
B
in
Q
Q
Q
. Prove that
Q
O
1
QO_1
Q
O
1
passes through the midpoint of
B
C
BC
BC
.
geometry