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National and Regional Contests
Bangladesh Contests
Bangladesh Mathematical Olympiad
2022 Bangladesh Mathematical Olympiad
5
5
Part of
2022 Bangladesh Mathematical Olympiad
Problems
(1)
Four points lie on the same circle
Source: BdMO 2022 Secondary P5
4/12/2022
In an acute triangle
△
A
B
C
\triangle ABC
△
A
BC
, the midpoint of
B
C
BC
BC
is
M
M
M
. Perpendicular lines
B
E
BE
BE
and
C
F
CF
CF
are drawn respectively on
A
C
AC
A
C
from
B
B
B
and on
A
B
AB
A
B
from
C
C
C
such that
E
E
E
and
F
F
F
lie on
A
C
AC
A
C
and
A
B
AB
A
B
respectively. The midpoint of
E
F
EF
EF
is
N
.
N.
N
.
M
N
MN
MN
intersects
A
B
AB
A
B
at
K
.
K.
K
.
Prove that, the four points
B
,
K
,
E
,
M
B,K,E,M
B
,
K
,
E
,
M
lie on the same circle.
geometry