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International Contests
Middle European Mathematical Olympiad
2017 Middle European Mathematical Olympiad
5
5
Part of
2017 Middle European Mathematical Olympiad
Problems
(1)
Midpoint of the segment joining two excentres of a triangle
Source: MEMO 2017 T5
8/25/2017
Let
A
B
C
ABC
A
BC
be an acute-angled triangle with
A
B
>
A
C
AB > AC
A
B
>
A
C
and circumcircle
Γ
\Gamma
Γ
. Let
M
M
M
be the midpoint of the shorter arc
B
C
BC
BC
of
Γ
\Gamma
Γ
, and let
D
D
D
be the intersection of the rays
A
C
AC
A
C
and
B
M
BM
BM
. Let
E
≠
C
E \neq C
E
=
C
be the intersection of the internal bisector of the angle
A
C
B
ACB
A
CB
and the circumcircle of the triangle
B
D
C
BDC
B
D
C
. Let us assume that
E
E
E
is inside the triangle
A
B
C
ABC
A
BC
and there is an intersection
N
N
N
of the line
D
E
DE
D
E
and the circle
Γ
\Gamma
Γ
such that
E
E
E
is the midpoint of the segment
D
N
DN
D
N
. Show that
N
N
N
is the midpoint of the segment
I
B
I
C
I_B I_C
I
B
I
C
, where
I
B
I_B
I
B
and
I
C
I_C
I
C
are the excentres of
A
B
C
ABC
A
BC
opposite to
B
B
B
and
C
C
C
, respectively.
geometry