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China Contests
China Team Selection Test
2023 China Team Selection Test
P16
P16
Part of
2023 China Team Selection Test
Problems
(1)
Orz Orz Orz Orz Orz Geometry III, and also orz Haozhe
Source: China TST 2023 Problem 16
3/29/2023
Let
Γ
,
Γ
1
,
Γ
2
\Gamma, \Gamma_1, \Gamma_2
Γ
,
Γ
1
,
Γ
2
be mutually tangent circles. The three circles are also tangent to a line
l
l
l
. Let
Γ
,
Γ
1
\Gamma, \Gamma_1
Γ
,
Γ
1
be tangent to each other at
B
1
B_1
B
1
,
Γ
,
Γ
2
\Gamma, \Gamma_2
Γ
,
Γ
2
be tangent to each other at
B
2
B_2
B
2
,
Γ
1
,
Γ
2
\Gamma_1, \Gamma_2
Γ
1
,
Γ
2
be tangent to each other at
C
C
C
.
Γ
,
Γ
1
,
Γ
2
\Gamma, \Gamma_1, \Gamma_2
Γ
,
Γ
1
,
Γ
2
are tangent to
l
l
l
at
A
,
A
1
,
A
2
A, A_1, A_2
A
,
A
1
,
A
2
respectively, where
A
A
A
is between
A
1
,
A
2
A_1,A_2
A
1
,
A
2
. Let
D
1
=
A
1
C
∩
A
2
B
2
,
D
2
=
A
2
C
∩
A
1
B
1
D_1 = A_1C \cap A_2B_2, D_2 = A_2C \cap A_1B_1
D
1
=
A
1
C
∩
A
2
B
2
,
D
2
=
A
2
C
∩
A
1
B
1
. Prove that
D
1
D
2
D_1D_2
D
1
D
2
is parallel to
l
l
l
.
geometry