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Turkey Contests
Turkey Team Selection Test
2020 Turkey Team Selection Test
7
Parallel chords
Parallel chords
Source: 2020 Turkey TST P7
March 14, 2020
geometry
Problem Statement
A
1
,
A
2
,
B
1
,
B
2
,
C
1
,
C
2
A_1,A_2,B_1,B_2,C_1,C_2
A
1
,
A
2
,
B
1
,
B
2
,
C
1
,
C
2
are points on a circle such that
A
1
A
2
∥
B
1
B
2
∥
C
1
C
2
A_1A_2 \parallel B_1B_2 \parallel C_1C_2
A
1
A
2
∥
B
1
B
2
∥
C
1
C
2
.
M
M
M
is a point on same circle
M
A
1
MA_1
M
A
1
and
B
2
C
2
B_2C_2
B
2
C
2
intersect at
X
X
X
,
M
B
1
MB_1
M
B
1
and
A
2
C
2
A_2C_2
A
2
C
2
intersect at
Y
Y
Y
,
M
C
1
MC_1
M
C
1
and
A
2
B
2
A_2B_2
A
2
B
2
intersect at
Z
Z
Z
.Prove that
X
,
Y
,
Z
X , Y ,Z
X
,
Y
,
Z
are collinear.
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