MathDB
Problems
Contests
International Contests
Middle European Mathematical Olympiad
2018 Middle European Mathematical Olympiad
5
5
Part of
2018 Middle European Mathematical Olympiad
Problems
(1)
DHMO is a parallelogram
Source: MEMO 2018 T5
9/3/2018
Let
A
B
C
ABC
A
BC
be an acute-angled triangle with
A
B
<
A
C
,
AB<AC,
A
B
<
A
C
,
and let
D
D
D
be the foot of its altitude from
A
,
A,
A
,
points
B
′
B'
B
′
and
C
′
C'
C
′
lie on the rays
A
B
AB
A
B
and
A
C
,
AC,
A
C
,
respectively , so that points
B
′
,
B',
B
′
,
C
′
C'
C
′
and
D
D
D
are collinear and points
B
,
B,
B
,
C
,
C,
C
,
B
′
B'
B
′
and
C
′
C'
C
′
lie on one circle with center
O
.
O.
O
.
Prove that if
M
M
M
is the midpoint of
B
C
BC
BC
and
H
H
H
is the orthocenter of
A
B
C
,
ABC,
A
BC
,
then
D
H
M
O
DHMO
DH
MO
is a parallelogram.
geometry
parallelogram
OHM