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Sharygin Geometry Olympiad
2010 Sharygin Geometry Olympiad
23
Prove that the lines AD, BE and CF are concurrent (23)
Prove that the lines AD, BE and CF are concurrent (23)
Source:
October 29, 2010
geometry proposed
geometry
Problem Statement
A cyclic hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
is such that
A
B
⋅
C
F
=
2
B
C
⋅
F
A
,
C
D
⋅
E
B
=
2
D
E
⋅
B
C
AB \cdot CF= 2BC \cdot FA, CD \cdot EB = 2 DE \cdot BC
A
B
⋅
CF
=
2
BC
⋅
F
A
,
C
D
⋅
EB
=
2
D
E
⋅
BC
and
E
F
⋅
A
D
=
2
F
A
⋅
D
E
.
EF \cdot AD = 2FA \cdot DE.
EF
⋅
A
D
=
2
F
A
⋅
D
E
.
Prove that the lines
A
D
,
B
E
AD, BE
A
D
,
BE
and
C
F
CF
CF
are concurrent.
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