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National and Regional Contests
Russia Contests
Sharygin Geometry Olympiad
2022 Sharygin Geometry Olympiad
12
12
Part of
2022 Sharygin Geometry Olympiad
Problems
(1)
Questionable parallelogram
Source: Sharygin CR 2022 P12
3/4/2022
Let
K
K
K
,
L
L
L
,
M
M
M
,
N
N
N
be the midpoints of sides
B
C
BC
BC
,
C
D
CD
C
D
,
D
A
DA
D
A
,
A
B
AB
A
B
respectively of a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
. The common points of segments
A
K
AK
A
K
,
B
L
BL
B
L
,
C
M
CM
CM
,
D
N
DN
D
N
divide each of them into three parts. It is known that the ratio of the length of the medial part to the length of the whole segment is the same for all segments. Does this yield that
A
B
C
D
ABCD
A
BC
D
is a parallelogram?
geometry
parallelogram