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Baltic Way
2001 Baltic Way
7
7
Part of
2001 Baltic Way
Problems
(1)
Circle through A of parallelogram ABCD
Source: Baltic Way 2001
11/17/2010
Given a parallelogram
A
B
C
D
ABCD
A
BC
D
. A circle passing through
A
A
A
meets the line segments
A
B
,
A
C
AB, AC
A
B
,
A
C
and
A
D
AD
A
D
at inner points
M
,
K
,
N
M,K,N
M
,
K
,
N
, respectively. Prove that
∣
A
B
∣
⋅
∣
A
M
∣
+
∣
A
D
∣
⋅
∣
A
N
∣
=
∣
A
K
∣
⋅
∣
A
C
∣
|AB|\cdot |AM | + |AD|\cdot |AN|=|AK|\cdot |AC|
∣
A
B
∣
⋅
∣
A
M
∣
+
∣
A
D
∣
⋅
∣
A
N
∣
=
∣
A
K
∣
⋅
∣
A
C
∣
geometry
parallelogram
geometry proposed