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Turkey Team Selection Test
1990 Turkey Team Selection Test
4
That area problem
That area problem
Source: Turkey TST 1990 - P4
September 11, 2013
geometry
analytic geometry
linear algebra
matrix
rotation
geometry proposed
Problem Statement
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral such that
E
,
F
∈
[
A
B
]
,
A
E
=
E
F
=
F
B
G
,
H
∈
[
B
C
]
,
B
G
=
G
H
=
H
C
K
,
L
∈
[
C
D
]
,
C
K
=
K
L
=
L
D
M
,
N
∈
[
D
A
]
,
D
M
=
M
N
=
N
A
\begin{array}{rl} E,F \in [AB],& AE = EF = FB \\ G,H \in [BC],& BG = GH = HC \\ K,L \in [CD],& CK = KL = LD \\ M,N \in [DA],& DM = MN = NA \end{array}
E
,
F
∈
[
A
B
]
,
G
,
H
∈
[
BC
]
,
K
,
L
∈
[
C
D
]
,
M
,
N
∈
[
D
A
]
,
A
E
=
EF
=
FB
BG
=
G
H
=
H
C
C
K
=
K
L
=
L
D
D
M
=
MN
=
N
A
Let
[
N
G
]
∩
[
L
E
]
=
{
P
}
,
[
N
G
]
∩
[
K
F
]
=
{
Q
}
,
[NG] \cap [LE] = \{P\}, [NG]\cap [KF] = \{Q\},
[
NG
]
∩
[
L
E
]
=
{
P
}
,
[
NG
]
∩
[
K
F
]
=
{
Q
}
,
[
M
H
]
∩
[
K
F
]
=
{
R
}
,
[
M
H
]
∩
[
L
E
]
=
{
S
}
{[}MH] \cap [KF] = \{R\}, [MH]\cap [LE]=\{S\}
[
M
H
]
∩
[
K
F
]
=
{
R
}
,
[
M
H
]
∩
[
L
E
]
=
{
S
}
Prove that [*]
A
r
e
a
(
A
B
C
D
)
=
9
⋅
A
r
e
a
(
P
Q
R
S
)
Area(ABCD) = 9 \cdot Area(PQRS)
A
re
a
(
A
BC
D
)
=
9
⋅
A
re
a
(
PQRS
)
[*]
N
P
=
P
Q
=
Q
G
NP=PQ=QG
NP
=
PQ
=
QG
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