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Contests
National and Regional Contests
Ukraine Contests
Official Ukraine Selection Cycle
Ukraine National Mathematical Olympiad
2021 Ukraine National Mathematical Olympiad
9.6
9.6
Part of
2021 Ukraine National Mathematical Olympiad
Problems
(1)
center of circle tangent internally to 2 intersecting circles, lies on bisector
Source: 2021 Ukraine NMO 9.6 10.6
4/4/2021
Circles
w
1
w_1
w
1
and
w
2
w_2
w
2
intersect at points
P
P
P
and
Q
Q
Q
and touch a circle
w
w
w
with center at point
O
O
O
internally at points
A
A
A
and
B
B
B
, respectively. It is known that the points
A
,
B
A,B
A
,
B
and
Q
Q
Q
lie on one line. Prove that the point
O
O
O
lies on the external bisector
∠
A
P
B
\angle APB
∠
A
PB
.(Nazar Serdyuk)
geometry
tangent circles