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2016 IMO Shortlist
G6
G6
Part of
2016 IMO Shortlist
Problems
(1)
Perpendicular to quadrilateral diagonal
Source: 2016 IMO Shortlist G6
7/19/2017
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral with
∠
A
B
C
=
∠
A
D
C
<
9
0
∘
\angle ABC = \angle ADC < 90^{\circ}
∠
A
BC
=
∠
A
D
C
<
9
0
∘
. The internal angle bisectors of
∠
A
B
C
\angle ABC
∠
A
BC
and
∠
A
D
C
\angle ADC
∠
A
D
C
meet
A
C
AC
A
C
at
E
E
E
and
F
F
F
respectively, and meet each other at point
P
P
P
. Let
M
M
M
be the midpoint of
A
C
AC
A
C
and let
ω
\omega
ω
be the circumcircle of triangle
B
P
D
BPD
BP
D
. Segments
B
M
BM
BM
and
D
M
DM
D
M
intersect
ω
\omega
ω
again at
X
X
X
and
Y
Y
Y
respectively. Denote by
Q
Q
Q
the intersection point of lines
X
E
XE
XE
and
Y
F
YF
Y
F
. Prove that
P
Q
⊥
A
C
PQ \perp AC
PQ
⊥
A
C
.
geometry
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