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Contests
National and Regional Contests
Poland Contests
Poland - Second Round
2014 Poland - Second Round
5.
5.
Part of
2014 Poland - Second Round
Problems
(1)
Tangency and circles
Source: Poland 2014 - Second Round P5
7/28/2019
Circles
o
1
o_1
o
1
and
o
2
o_2
o
2
tangent to some line at points
A
A
A
and
B
B
B
, respectively, intersect at points
X
X
X
and
Y
Y
Y
(
X
X
X
is closer to the line
A
B
AB
A
B
). Line
A
X
AX
A
X
intersects
o
2
o_2
o
2
at point
P
≠
X
P\neq X
P
=
X
. Tangent to
o
2
o_2
o
2
at point
P
P
P
intersects line
A
B
AB
A
B
at point
Q
Q
Q
. Prove that
∢
X
Y
B
=
∢
B
Y
Q
\sphericalangle XYB = \sphericalangle BYQ
∢
X
Y
B
=
∢
B
Y
Q
.
geometry
circles