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National and Regional Contests
Poland Contests
Poland - Second Round
2024 Poland - Second Round
2
2
Part of
2024 Poland - Second Round
Problems
(1)
Geo with angle and length conditions
Source: Polish MO Second round 2024 P2
2/9/2024
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral with
∠
A
B
C
=
∠
A
D
C
=
12
0
∘
\angle ABC=\angle ADC=120^{\circ}
∠
A
BC
=
∠
A
D
C
=
12
0
∘
. The point
E
E
E
lies on the segment
A
D
AD
A
D
and is such that
A
E
⋅
B
C
=
A
B
⋅
D
E
AE \cdot BC=AB \cdot DE
A
E
⋅
BC
=
A
B
⋅
D
E
and similarly the point
F
F
F
lies on the segment
B
C
BC
BC
and satisfies
B
F
⋅
C
D
=
A
D
⋅
C
F
BF \cdot CD=AD \cdot CF
BF
⋅
C
D
=
A
D
⋅
CF
. Show that
B
E
BE
BE
and
D
F
DF
D
F
are parallel.
geometry