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Sharygin Geometry Olympiad
2019 Sharygin Geometry Olympiad
18
18
Part of
2019 Sharygin Geometry Olympiad
Problems
(1)
Sharygin CR 2019 P18
Source: Sharygin CR 2019 P18 (Grade 10 - 11)
3/6/2019
A quadrilateral
A
B
C
D
ABCD
A
BC
D
without parallel sidelines is circumscribed around a circle centered at
I
I
I
. Let
K
,
L
,
M
K, L, M
K
,
L
,
M
and
N
N
N
be the midpoints of
A
B
,
B
C
,
C
D
AB, BC, CD
A
B
,
BC
,
C
D
and
D
A
DA
D
A
respectively. It is known that
A
B
⋅
C
D
=
4
I
K
⋅
I
M
AB \cdot CD = 4IK \cdot IM
A
B
⋅
C
D
=
4
I
K
⋅
I
M
. Prove that
B
C
⋅
A
D
=
4
I
L
⋅
I
N
BC \cdot AD = 4IL \cdot IN
BC
⋅
A
D
=
4
I
L
⋅
I
N
.
geometry