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National and Regional Contests
Turkey Contests
Turkey Team Selection Test
2017 Turkey Team Selection Test
3
3
Part of
2017 Turkey Team Selection Test
Problems
(1)
Geometry
Source: Turkey Tst 2017 p3
3/30/2017
At the
A
B
C
ABC
A
BC
triangle the midpoints of
B
C
,
A
C
,
A
B
BC, AC, AB
BC
,
A
C
,
A
B
are respectively
D
,
E
,
F
D, E, F
D
,
E
,
F
and the triangle tangent to the incircle at
G
G
G
,
H
H
H
and
I
I
I
in the same order.The midpoint of
A
D
AD
A
D
is
J
J
J
.
B
J
BJ
B
J
and
A
G
AG
A
G
intersect at point
K
K
K
. The
C
ā
C-
C
ā
centered circle passing through
A
A
A
cuts the
[
C
B
[CB
[
CB
ray at point
X
X
X
. The line passing through
K
K
K
and parallel to the
B
C
BC
BC
and
A
X
AX
A
X
meet at
U
U
U
.
I
U
IU
I
U
and
B
C
BC
BC
intersect at the
P
P
P
point. There is
Y
Y
Y
point chosen at incircle.
P
Y
PY
P
Y
is tangent to incircle at point
Y
Y
Y
. Prove that
D
,
E
,
F
,
Y
D, E, F, Y
D
,
E
,
F
,
Y
are cyclic.
circles
incenter
geometry