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National and Regional Contests
Greece Contests
Greece National Olympiad
2018 Greece National Olympiad
2
2
Part of
2018 Greece National Olympiad
Problems
(1)
Collinear if and only if
Source: 2018 Greece National Olympiad Problem 2
3/3/2018
Let
A
B
C
ABC
A
BC
be an acute-angled triangle with
A
B
<
A
C
<
B
C
AB<AC<BC
A
B
<
A
C
<
BC
and
c
(
O
,
R
)
c(O,R)
c
(
O
,
R
)
the circumscribed circle. Let
D
,
E
D, E
D
,
E
be points in the small arcs
A
C
,
A
B
AC, AB
A
C
,
A
B
respectively. Let
K
K
K
be the intersection point of
B
D
,
C
E
BD,CE
B
D
,
CE
and
N
N
N
the second common point of the circumscribed circles of the triangles
B
K
E
BKE
B
K
E
and
C
K
D
CKD
C
KD
. Prove that
A
,
K
,
N
A, K, N
A
,
K
,
N
are collinear if and only if
K
K
K
belongs to the symmedian of
A
B
C
ABC
A
BC
passing from
A
A
A
.
geometry
symmedian