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Balkan MO Shortlist
2020 Balkan MO Shortlist
G5
G5
Part of
2020 Balkan MO Shortlist
Problems
(1)
concurrency wanted, starting with a 45-135/2 - 135/2 triangle
Source: 2020 Balkan MO shortlist G5
9/14/2021
Let
A
B
C
ABC
A
BC
be an isosceles triangle with
A
B
=
A
C
AB = AC
A
B
=
A
C
and
∠
A
=
4
5
o
\angle A = 45^o
∠
A
=
4
5
o
. Its circumcircle
(
c
)
(c)
(
c
)
has center
O
,
M
O, M
O
,
M
is the midpoint of
B
C
BC
BC
and
D
D
D
is the foot of the perpendicular from
C
C
C
to
A
B
AB
A
B
. With center
C
C
C
and radius
C
D
CD
C
D
we draw a circle which internally intersects
A
C
AC
A
C
at the point
F
F
F
and the circle
(
c
)
(c)
(
c
)
at the points
Z
Z
Z
and
E
E
E
, such that
Z
Z
Z
lies on the small arc
B
C
BC
BC
and
E
E
E
on the small arc
A
C
AC
A
C
. Prove that the lines
Z
E
ZE
ZE
,
C
O
CO
CO
,
F
M
FM
FM
are concurrent.Brazitikos Silouanos, Greece
geometry
circumcircle