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International Contests
Nordic
2017 Nordic
3
3
Part of
2017 Nordic
Problems
(1)
Collinearity in isosceles trapezoid
Source: Nordic Mathematical Contest 2017 Problem 3
4/3/2017
Let
M
M
M
and
N
N
N
be the midpoints of the sides
A
C
AC
A
C
and
A
B
AB
A
B
, respectively, of an acute triangle
A
B
C
ABC
A
BC
,
A
B
≠
A
C
AB \neq AC
A
B
=
A
C
. Let
ω
B
\omega_B
ω
B
be the circle centered at
M
M
M
passing through
B
B
B
, and let
ω
C
\omega_C
ω
C
be the circle centered at
N
N
N
passing through
C
C
C
. Let the point
D
D
D
be such that
A
B
C
D
ABCD
A
BC
D
is an isosceles trapezoid with
A
D
AD
A
D
parallel to
B
C
BC
BC
. Assume that
ω
B
\omega_B
ω
B
and
ω
C
\omega_C
ω
C
intersect in two distinct points
P
P
P
and
Q
Q
Q
. Show that
D
D
D
lies on the line
P
Q
PQ
PQ
.
geometry