MathDB
Problems
Contests
National and Regional Contests
Moldova Contests
EGMO TST - Moldova
2018 Moldova EGMO TST
3
3
Part of
2018 Moldova EGMO TST
Problems
(1)
Geometry problem
Source: EGMO 2018 Moldova TST
2/11/2018
Let
△
A
B
C
\triangle ABC
△
A
BC
be an acute triangle.
O
O
O
denote its circumcenter.Points
D
D
D
,
E
E
E
,
F
F
F
are the midpoints of the sides
B
C
BC
BC
,
C
A
CA
C
A
,and
A
B
AB
A
B
.Let
M
M
M
be a point on the side
B
C
BC
BC
.
A
M
∩
E
F
=
{
N
}
AM \cap EF = \big\{ N \big\}
A
M
∩
EF
=
{
N
}
.
O
N
∩
(
O
D
M
)
=
{
P
}
ON \cap \big( ODM \big) = \big\{ P \big\}
ON
∩
(
O
D
M
)
=
{
P
}
Prove that
M
′
M'
M
′
lie on
(
D
E
F
)
\big(DEF\big)
(
D
EF
)
where
M
′
M'
M
′
is the symmetrical point of
M
M
M
thought the midpoint of
D
P
DP
D
P
.
geometry