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CentroAmerican
2006 CentroAmerican
6
6
Part of
2006 CentroAmerican
Problems
(1)
Product of ratios
Source: Central American Olympiad 2006, Problem 6
4/30/2007
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral.
I
=
A
C
∩
B
D
I=AC\cap BD
I
=
A
C
∩
B
D
, and
E
E
E
,
H
H
H
,
F
F
F
and
G
G
G
are points on
A
B
AB
A
B
,
B
C
BC
BC
,
C
D
CD
C
D
and
D
A
DA
D
A
respectively, such that
E
F
∩
G
H
=
I
EF \cap GH= I
EF
∩
G
H
=
I
. If
M
=
E
G
∩
A
C
M=EG \cap AC
M
=
EG
∩
A
C
,
N
=
H
F
∩
A
C
N=HF \cap AC
N
=
H
F
∩
A
C
, show that
A
M
I
M
⋅
I
N
C
N
=
I
A
I
C
.
\frac{AM}{IM}\cdot \frac{IN}{CN}=\frac{IA}{IC}.
I
M
A
M
⋅
CN
I
N
=
I
C
I
A
.
ratio
trigonometry
projective geometry
geometry proposed
geometry