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Contests
National and Regional Contests
Turkey Contests
Turkey Team Selection Test
2016 Turkey Team Selection Test
6
6
Part of
2016 Turkey Team Selection Test
Problems
(1)
Concyclic Points in an Isosceles Triangle
Source: Turkey TST 2016 P6
4/10/2016
In a triangle
A
B
C
ABC
A
BC
with
A
B
=
A
C
AB=AC
A
B
=
A
C
, let
D
D
D
be the midpoint of
[
B
C
]
[BC]
[
BC
]
. A line passing through
D
D
D
intersects
A
B
AB
A
B
at
K
K
K
,
A
C
AC
A
C
at
L
L
L
. A point
E
E
E
on
[
B
C
]
[BC]
[
BC
]
different from
D
D
D
, and a point
P
P
P
on
A
E
AE
A
E
is taken such that
∠
K
P
L
=
9
0
∘
−
1
2
∠
K
A
L
\angle KPL=90^\circ-\frac{1}{2}\angle KAL
∠
K
P
L
=
9
0
∘
−
2
1
∠
K
A
L
and
E
E
E
lies between
A
A
A
and
P
P
P
. The circumcircle of triangle
P
D
E
PDE
P
D
E
intersects
P
K
PK
P
K
at point
X
X
X
,
P
L
PL
P
L
at point
Y
Y
Y
for the second time. Lines
D
X
DX
D
X
and
A
B
AB
A
B
intersect at
M
M
M
, and lines
D
Y
DY
D
Y
and
A
C
AC
A
C
intersect at
N
N
N
. Prove that the points
P
,
M
,
A
,
N
P,M,A,N
P
,
M
,
A
,
N
are concyclic.
geometry