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International Contests
Balkan MO
2005 Balkan MO
1
1
Part of
2005 Balkan MO
Problems
(1)
Balkan 2005-1
Source:
5/6/2005
Let
A
B
C
ABC
A
BC
be an acute-angled triangle whose inscribed circle touches
A
B
AB
A
B
and
A
C
AC
A
C
at
D
D
D
and
E
E
E
respectively. Let
X
X
X
and
Y
Y
Y
be the points of intersection of the bisectors of the angles
∠
A
C
B
\angle ACB
∠
A
CB
and
∠
A
B
C
\angle ABC
∠
A
BC
with the line
D
E
DE
D
E
and let
Z
Z
Z
be the midpoint of
B
C
BC
BC
. Prove that the triangle
X
Y
Z
XYZ
X
Y
Z
is equilateral if and only if
∠
A
=
6
0
∘
\angle A = 60^\circ
∠
A
=
6
0
∘
.
geometry
incenter