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JBMO ShortLists
2019 JBMO Shortlist
G6
G6
Part of
2019 JBMO Shortlist
Problems
(1)
JBMO Shortlist 2019 G6
Source:
9/12/2020
Let
A
B
C
ABC
A
BC
be a non-isosceles triangle with incenter
I
I
I
. Let
D
D
D
be a point on the segment
B
C
BC
BC
such that the circumcircle of
B
I
D
BID
B
I
D
intersects the segment
A
B
AB
A
B
at
E
≠
B
E\neq B
E
=
B
, and the circumcircle of
C
I
D
CID
C
I
D
intersects the segment
A
C
AC
A
C
at
F
≠
C
F\neq C
F
=
C
. The circumcircle of
D
E
F
DEF
D
EF
intersects
A
B
AB
A
B
and
A
C
AC
A
C
at the second points
M
M
M
and
N
N
N
respectively. Let
P
P
P
be the point of intersection of
I
B
IB
I
B
and
D
E
DE
D
E
, and let
Q
Q
Q
be the point of intersection of
I
C
IC
I
C
and
D
F
DF
D
F
. Prove that the three lines
E
N
,
F
M
EN, FM
EN
,
FM
and
P
Q
PQ
PQ
are parallel.Proposed by Saudi Arabia
geometry
incenter