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Kvant 2019
M2571
M2571
Part of
Kvant 2019
Problems
(1)
Cute trapezoid geometry
Source: Kvant Magazine No. 8 2019 M2571
3/14/2023
Let
A
B
C
D
ABCD
A
BC
D
be a trapezoid with
A
D
∥
B
C
AD \parallel BC
A
D
∥
BC
,
A
D
<
B
C
AD < BC
A
D
<
BC
. Let
E
E
E
be a point on the side
A
B
AB
A
B
and
F
F
F
be point on the side
C
D
CD
C
D
. The circle
(
A
E
F
)
(AEF)
(
A
EF
)
intersects the segment
A
D
AD
A
D
again at
A
1
A_1
A
1
and the circle
(
C
E
F
)
(CEF)
(
CEF
)
intersects these segment
B
C
BC
BC
again at
C
1
C_1
C
1
. Prove that the lines
A
1
C
1
A_1 C_1
A
1
C
1
,
B
D
BD
B
D
and
E
F
EF
EF
are concurrent.Proposed by A. Kuznetsov
geometry
trapezoid
Kvant