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239 Open Math Olympiad
2022 239 Open Mathematical Olympiad
7
7
Part of
2022 239 Open Mathematical Olympiad
Problems
(1)
Problem 7
Source: 239-School Open Olympiad (Senior Level)
4/25/2022
Points
A
,
B
,
C
A,B,C
A
,
B
,
C
are chosen inside the triangle
A
1
B
1
C
1
,
A_{1}B_{1}C_{1},
A
1
B
1
C
1
,
so that the quadrilaterals
B
1
C
B
C
1
,
C
1
A
C
A
1
B_{1}CBC_{1}, C_{1}ACA_{1}
B
1
CB
C
1
,
C
1
A
C
A
1
and
A
1
B
A
B
1
A_{1}BAB_{1}
A
1
B
A
B
1
are inscribed in the circles
Ω
A
,
Ω
B
\Omega _{A}, \Omega _{B}
Ω
A
,
Ω
B
and
Ω
C
,
\Omega _{C},
Ω
C
,
respectively. The circle
Y
A
Y_{A}
Y
A
internally touches the circles
Ω
B
,
Ω
C
\Omega _{B}, \Omega _{C}
Ω
B
,
Ω
C
and externally touches the circle
Ω
A
.
\Omega _{A}.
Ω
A
.
The common interior tangent to the circles
Y
A
Y_{A}
Y
A
and
Ω
A
\Omega _{A}
Ω
A
intersects the line
B
C
BC
BC
at point
A
′
.
A'.
A
′
.
Points
B
′
B'
B
′
and
C
′
C'
C
′
are analogously defined. Prove that points
A
′
,
B
′
A',B'
A
′
,
B
′
and
C
′
C'
C
′
are lying on the same line.
geometry
tangent