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Contests
National and Regional Contests
India Contests
India National Olympiad
2011 India National Olympiad
5
5
Part of
2011 India National Olympiad
Problems
(1)
Cyclic quadrilateral with midpoints of arcs;-[INMO 2011]
Source:
2/6/2011
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral inscribed in a circle
Γ
.
\Gamma.
Γ.
Let
E
,
F
,
G
,
H
E,F,G,H
E
,
F
,
G
,
H
be the midpoints of arcs
A
B
,
B
C
,
C
D
,
A
D
AB,BC,CD,AD
A
B
,
BC
,
C
D
,
A
D
of
Γ
,
\Gamma,
Γ
,
respectively. Suppose that
A
C
⋅
B
D
=
E
G
⋅
F
H
.
AC\cdot BD=EG\cdot FH.
A
C
⋅
B
D
=
EG
⋅
F
H
.
Show that
A
C
,
B
D
,
E
G
,
F
H
AC,BD,EG,FH
A
C
,
B
D
,
EG
,
F
H
are all concurrent.
geometry
cyclic quadrilateral
geometry unsolved