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Czech-Polish-Slovak Match
2022 Czech-Austrian-Polish-Slovak Match
3
3
Part of
2022 Czech-Austrian-Polish-Slovak Match
Problems
(1)
Geo with two circles and a variable line
Source: Czech-Polish-Slovak Match 2022 P3
9/3/2022
Circles
Ω
1
\Omega_1
Ω
1
and
Ω
2
\Omega_2
Ω
2
with different radii intersect at two points, denote one of them by
P
P
P
. A variable line
l
l
l
passing through
P
P
P
intersects the arc of
Ω
1
\Omega_1
Ω
1
which is outside of
Ω
2
\Omega_2
Ω
2
at
X
1
X_1
X
1
, and the arc of
Ω
2
\Omega_2
Ω
2
which is outside of
Ω
1
\Omega_1
Ω
1
at
X
2
X_2
X
2
. Let
R
R
R
be the point on segment
X
1
X
2
X_1X_2
X
1
X
2
such that
X
1
P
=
R
X
2
X_1P = RX_2
X
1
P
=
R
X
2
. The tangent to
Ω
1
\Omega_1
Ω
1
through
X
1
X_1
X
1
meets the tangent to
Ω
2
\Omega_2
Ω
2
through
X
2
X_2
X
2
at
T
T
T
. Prove that line
R
T
RT
RT
/is tangent to a fixed circle, independent of the choice of
l
l
l
.
geometry